Scalar problems in junctions of rods and a plate.
II. Self-adjoint extensions and simulation models.

R.Bunoiu, G.Cardone, S.A.Nazarov University of Lorraine, IECL, CNRS UMR 7502, 3 rue Augustin Fresnel, 57073, Metz, France; email: renata.bunoiu@univ-lorraine.fr.Università del Sannio, Department of Engineering, Corso Garibaldi, 107, 82100 Benevento, Italy; email: giuseppe.cardone@unisannio.itMathematics and Mechanics Faculty, St. Petersburg State University, 198504, Universitetsky pr., 28, Stary Peterhof, Russia; Peter the Great St. Petersburg State Polytechnical University, Polytechnicheskaya ul., 29, St. Petersburg, 195251, Russia; Institute of Problems of Mechanical Engineering RAS, V.O., Bolshoj pr., 61, St. Petersburg, 199178, Russia; email: srgnazarov@yahoo.co.uk.
Abstract

In this work we deal with a scalar spectral mixed boundary value problem in a spacial junction of thin rods and a plate. Constructing asymptotics of the eigenvalues, we employ two equipollent asymptotic models posed on the skeleton of the junction, that is, a hybrid domain. We, first, use the technique of self-adjoint extensions and, second, we impose algebraic conditions at the junction points in order to compile a problem in a function space with detached asymptotics. The latter problem is involved into a symmetric generalized Green formula and, therefore, admits the variational formulation. In comparison with a primordial asymptotic procedure, these two models provide much better proximity of the spectra of the problems in the spacial junction and in its skeleton. However, they exhibit the negative spectrum of finite multiplicity and for these ”parasitic” eigenvalues we derive asymptotic formulas to demonstrate that they do not belong to the service area of the developed asymptotic models.

Keywords: junction of thin rods and plate, scalar spectral problem, asymptotics, dimension reduction, self-adjoint extensions of differential operators, function space with detached asymptotics

MSC: 35B40, 35C20, 74K30.

1 Introduction

1.1 Motivations

As it was observed in [4], an asymptotic expansion of a solution of the Poisson scalar mixed boundary-value problem in a junction of thin rods and a thin plate in a certain range of physical parameters gains the rational dependence on the big parameter |lnh||\ln h| where h>00h>0 is a small parameter characterizing the diameters of the rods and the thickness of the plate. Similar asymptotic forms had been discovered for other elliptic problems stated in domains with singular perturbations of the boundaries, see the books [16, 11, 9]. The aim of this paper is to study the scalar spectral problem associated to the Poisson problem studied in [4].

Based on the asymptotic procedure in [16, §2.2.4, §5.5.2, §9.1.3] and [17, 4], it is quite predictable that the asymptotic expansions of the eigenpairs ”eigenvalue/eigenfunction” become much more complicated than the one obtained for the solution of the Poisson problem and purchase the holomorphic dependence on the parameter |lnh|1superscript1|\ln h|^{-1}. Such sophistication of the asymptotic expansions and the lack of algorithms allowing to clarify the appearing holomorphic functions make them almost useless in applications, especially for combined numerical and asymptotic methods.

In [14] J.-L. Lions announced as an open question an application of the technique of self-adjoint extensions of differential operators for modeling boundary-value problems with singular perturbations. This technique has been employed in [18, 19, 20] and others to deal with particular types of perturbations but at our knowledge, was never used before for the study of spectral problems for junctions of thin domains with different limit dimension like 1d and 2d for the rods and plate in our 3d junction.

The use of such techniques in the asymptotic analysis here is the main novelty of our work. We observe that to the spacial junction Ξ(h)Ξ\Xi(h) represented in fig. 1,a, corresponds the hybrid domain Ξ0superscriptΞ0\Xi^{0} depicted in fig. 2,a, which consists of several line segments joined to some interior points of a planar domain. Supplying these elements of Ξ0superscriptΞ0\Xi^{0} with differential structures, we describe a family of all self-adjoint operators which is parametrized by a finite set of free parameters and choose an appropriate set by examining the boundary layer phenomenon in the vicinity of the junction zones, namely where the rods are inserted into small sockets in the plate, fig. 1,b. As a result, we obtain a model which provides the satisfactory proximity O(h1/2(1+|lnh|)3)𝑂superscript12superscript13O(h^{1/2}(1+|\ln h|)^{3}) instead of the uncomfortable one O((1+|lnh|)2)𝑂superscript12O((1+|\ln h|)^{-2}) within a simplified but comprehensible version of the conventional asymptotic procedure. It should be mentioned that our model involves only one scalar integral characteristic of each junction zone.

Refer to caption
Figure 1: The junction (a) and its elements (b).
Refer to caption
Figure 2: The skeleton of the junction, the coupled (a) and disjoint (b) hybrid domain.

1.2 Formulation of the spectral problem

Given a small parameter h(0,h0]0subscript0h\in(0,h_{0}], we introduce the thin plate and rods

Ω0(h)subscriptΩ0\displaystyle\Omega_{0}\left(h\right) ={x=(y,z)3:y=(y1,y2)ω0, ζ:=h1z(0,1)},absentconditional-set𝑥𝑦𝑧superscript3formulae-sequence𝑦subscript𝑦1subscript𝑦2subscript𝜔0assign 𝜁superscript1𝑧01\displaystyle=\left\{x=\left(y,z\right)\in\mathbb{R}^{3}:y=(y_{1},y_{2})\in\mathbb{\omega}_{0},\text{ }\zeta:=h^{-1}z\in\left(0,1\right)\right\}, (1.1)
Ωj(h)subscriptΩ𝑗\displaystyle\Omega_{j}\left(h\right) ={x:ηj=h1(yPj)ωj, zIj:=(0,lj)},j=1,,J.formulae-sequenceabsentconditional-set𝑥formulae-sequencesuperscript𝜂𝑗superscript1𝑦superscript𝑃𝑗subscript𝜔𝑗 𝑧subscript𝐼𝑗assign0subscript𝑙𝑗𝑗1𝐽\displaystyle=\left\{x:\eta^{j}=h^{-1}\left(y-P^{j}\right)\in\mathbb{\omega}_{j},\text{ \ }z\in I_{j}:=(0,l_{j})\right\},\ j=1,...,J. (1.2)

Here, ωp,subscript𝜔𝑝\omega_{p}, p=0,1,,J,𝑝01𝐽p=0,1,...,J, are domains in the plane 2superscript2\mathbb{R}^{2} with smooth (for simplicity) boundaries ωpsubscript𝜔𝑝\partial\mathbb{\omega}_{p} and the compact closures ω¯p=ωpωp;subscript¯𝜔𝑝subscript𝜔𝑝subscript𝜔𝑝\overline{\mathbb{\omega}}_{p}=\mathbb{\omega}_{p}\cup\partial\mathbb{\omega}_{p}; l1,,lJsubscript𝑙1subscript𝑙𝐽l_{1},...,l_{J} are positive numbers independent of hh, and Pjω0superscript𝑃𝑗subscript𝜔0P^{j}\in\omega_{0}, PjPksuperscript𝑃𝑗superscript𝑃𝑘P^{j}\neq P^{k} for jk𝑗𝑘j\neq k. Reducing a characteristic size of ω0subscript𝜔0\mathbb{\omega}_{0} to 111, we make Cartesian coordinates and all geometric parameters dimensionless. We fix some h0(0,min{l1,,lJ})subscript00subscript𝑙1subscript𝑙𝐽h_{0}\in(0,\min\left\{l_{1},...,l_{J}\right\}) such that, for h(0,h0],0subscript0h\in\left(0,h_{0}\right], ω¯jhω0superscriptsubscript¯𝜔𝑗subscript𝜔0\overline{\mathbb{\omega}}_{j}^{h}\subset\mathbb{\omega}_{0} and ω¯jhω¯kh=,superscriptsubscript¯𝜔𝑗superscriptsubscript¯𝜔𝑘\overline{\mathbb{\omega}}_{j}^{h}\cap\overline{\mathbb{\omega}}_{k}^{h}=\emptyset, jk𝑗𝑘j\neq k, where ωjh={y:ηjωj}superscriptsubscript𝜔𝑗conditional-set𝑦superscript𝜂𝑗subscript𝜔𝑗\mathbb{\omega}_{j}^{h}=\left\{y:\eta^{j}\in\mathbb{\omega}_{j}\right\} is the cross section of the rod Ωj(h)subscriptΩ𝑗\Omega_{j}\left(h\right) while ωjh(0)superscriptsubscript𝜔𝑗0\mathbb{\omega}_{j}^{h}(0) and ωjh(lj)superscriptsubscript𝜔𝑗subscript𝑙𝑗\mathbb{\omega}_{j}^{h}(l_{j}) are its lower and upper ends. In the sequel the bound h0subscript0h_{0} can be diminished but always remains strictly positive.

The rods (1.2) are plugged into sockets, i.e., holes in the plate, fig. 1,b and a,,

Ω(h)=ω(h)×(0,h),ω(h)=ω0(ω¯1h..ω¯Jh),\Omega_{\bullet}\left(h\right)=\mathbb{\omega}_{\bullet}\left(h\right)\times\left(0,h\right),\ \ \ \mathbb{\omega}_{\bullet}\left(h\right)=\mathbb{\omega}_{0}\mathbb{\diagdown}\left(\overline{\mathbb{\omega}}_{1}^{h}\cup.....\cup\overline{\mathbb{\omega}}_{{J}}^{h}\right), (1.3)

and compose the junction

Ξ(h)=Ω(h)Ω1(h).ΩJ(h).formulae-sequenceΞsubscriptΩsubscriptΩ1subscriptΩ𝐽\Xi\left(h\right)=\Omega_{\bullet}\left(h\right)\cup\Omega_{1}\left(h\right)\cup....\cup\Omega_{J}\left(h\right). (1.4)

The lateral side of the plate (1.1) and (1.3) and their bases are denoted by υ0(h)=ω0×(0,h)subscript𝜐0subscript𝜔00\mathbb{\upsilon}_{0}\left(h\right)=\partial\omega_{0}\times(0,h) and

ς0i(h)={x:yω0,z=ih},ςi(h)={x:yω,z=ih},i=0,1.formulae-sequencesuperscriptsubscript𝜍0𝑖conditional-set𝑥formulae-sequence𝑦subscript𝜔0𝑧𝑖formulae-sequencesuperscriptsubscript𝜍𝑖conditional-set𝑥formulae-sequence𝑦subscript𝜔𝑧𝑖𝑖01\varsigma_{0}^{i}\left(h\right)=\left\{x:y\in\mathbb{\omega}_{0},\ z=ih\right\},\ \ \ \varsigma_{\bullet}^{i}\left(h\right)=\left\{x:y\in\mathbb{\omega}_{\bullet},\ z=ih\right\},\ \ i=0,1. (1.5)

The lateral side of the rod Ωj(h)subscriptΩ𝑗\Omega_{j}\left(h\right) is divided into two parts

Σj(h)=ωjh×(h,lj),υjh=ωjh×(0,h),formulae-sequencesubscriptΣ𝑗superscriptsubscript𝜔𝑗subscript𝑙𝑗superscriptsubscript𝜐𝑗superscriptsubscript𝜔𝑗0\Sigma_{j}\left(h\right)=\partial\mathbb{\omega}_{j}^{h}\times\left(h,l_{j}\right),\ \ \mathbb{\upsilon}_{j}^{h}=\partial\mathbb{\omega}_{j}^{h}\times\left(0,h\right),

where the latter is the junctional boundary of the socket.

In the junction (1.4) we consider a spectral problem consisting of the differential equations with the Laplace operator

Δxu0(h,x)subscriptΔ𝑥subscript𝑢0𝑥\displaystyle-\Delta_{x}u_{0}\left(h,x\right) =λ(h)u0(h,x),xΩ(h),formulae-sequenceabsent𝜆subscript𝑢0𝑥𝑥subscriptΩ\displaystyle=\lambda(h)u_{0}\left(h,x\right),\ \ \ x\in\Omega_{\bullet}\left(h\right), (1.6)
γj(h)Δxuj(h,x)subscript𝛾𝑗subscriptΔ𝑥subscript𝑢𝑗𝑥\displaystyle-\gamma_{j}\left(h\right)\Delta_{x}u_{j}\left(h,x\right) =λ(h)ρj(h)uj(h,x),xΩj(h),formulae-sequenceabsent𝜆subscript𝜌𝑗subscript𝑢𝑗𝑥𝑥subscriptΩ𝑗\displaystyle=\lambda(h)\rho_{j}(h)u_{j}\left(h,x\right),\ \ \ x\in\Omega_{j}\left(h\right), (1.7)

the Neumann and Dirichlet boundary conditions

νu0(h,x)subscript𝜈subscript𝑢0𝑥\displaystyle\partial_{\nu}u_{0}\left(h,x\right) =0,xΣ(h)=υ0(h)ς0(h)ς1(h),formulae-sequenceabsent0𝑥subscriptΣsubscript𝜐0superscriptsubscript𝜍0superscriptsubscript𝜍1\displaystyle=0,\ \ \ \ x\in\Sigma_{\bullet}\left(h\right)=\mathbb{\upsilon}_{0}\left(h\right)\cup\varsigma_{\bullet}^{0}\left(h\right)\cup\varsigma_{\bullet}^{1}\left(h\right), (1.8)
γj(h)νuj(h,x)subscript𝛾𝑗subscript𝜈subscript𝑢𝑗𝑥\displaystyle\gamma_{j}\left(h\right)\partial_{\nu}u_{j}\left(h,x\right) =0,xΣj(h)ωjh(0),formulae-sequenceabsent0𝑥subscriptΣ𝑗superscriptsubscript𝜔𝑗0\displaystyle=0,\ \ \ x\in\Sigma_{j}\left(h\right)\cup\mathbb{\omega}_{j}^{h}\left(0\right), (1.9)
uj(h,x)subscript𝑢𝑗𝑥\displaystyle u_{j}\left(h,x\right) =0,xωjh(lj),formulae-sequenceabsent0𝑥superscriptsubscript𝜔𝑗subscript𝑙𝑗\displaystyle=0,\ \ \ x\in\mathbb{\omega}_{j}^{h}\left(l_{j}\right), (1.10)

and the transmission conditions

u0(h,x)subscript𝑢0𝑥\displaystyle u_{0}\left(h,x\right) =uj(h,x),xυjh,formulae-sequenceabsentsubscript𝑢𝑗𝑥𝑥superscriptsubscript𝜐𝑗\displaystyle=u_{j}\left(h,x\right),\ \ \ \ x\in\mathbb{\upsilon}_{j}^{h}, (1.11)
νu0(h,x)subscript𝜈subscript𝑢0𝑥\displaystyle\partial_{\nu}u_{0}\left(h,x\right) =γj(h)νuj(h,x),xυjh.formulae-sequenceabsentsubscript𝛾𝑗subscript𝜈subscript𝑢𝑗𝑥𝑥superscriptsubscript𝜐𝑗\displaystyle=\gamma_{j}\left(h\right)\partial_{\nu}u_{j}\left(h,x\right),\ \ \ x\in\mathbb{\upsilon}_{j}^{h}. (1.12)

Here, λ(h)𝜆\lambda(h) is a spectral parameter, j=1,,J,u0𝑗1𝐽subscript𝑢0j=1,...,J,\ u_{0} and ujsubscript𝑢𝑗u_{j} are restrictions of the function u𝑢u on Ω(h)subscriptΩ\Omega_{\bullet}\left(h\right) and Ωj(h)subscriptΩ𝑗\Omega_{j}\left(h\right), respectively, and νsubscript𝜈\partial_{\nu} is the outward normal derivative at the surface of Ξ(h)¯¯Ξ\overline{\Xi(h)} in (1.8) and (1.9) while in the second transmission condition νsubscript𝜈\partial_{\nu} is outward with respect to the rods. In what follows we mainly deal with the coefficients

γj(h)=γjhα,γj>0,ρj(h)=ρjhα,ρj>0,formulae-sequencesubscript𝛾𝑗subscript𝛾𝑗superscript𝛼formulae-sequencesubscript𝛾𝑗0formulae-sequencesubscript𝜌𝑗subscript𝜌𝑗superscript𝛼subscript𝜌𝑗0\gamma_{j}\left(h\right)=\gamma_{j}h^{-\alpha},\ \gamma_{j}>0,\ \ \ \ \ \rho_{j}\left(h\right)=\rho_{j}h^{-\alpha},\ \rho_{j}>0, (1.13)

in the most informative but complicated case

α=1.𝛼1\alpha=1. (1.14)

We emphasize that in the case (1.14) the limit passage h+00h\rightarrow+0 in the problem (1.6)-(1.12) leads to the eigenvalue problem for a self-adjoint operator in the skeleton of (1.4) in fig. 2a, a hybrid domain, while, for α>1𝛼1\alpha>1 and α<1,𝛼1\alpha<1, this limit operator decouples, cf fig. 2b. We discuss the latter cases in Section 5 and pay attention to the homogeneous junction with

α=0,γj=ρj=1,j=1,,J.formulae-sequenceformulae-sequence𝛼0subscript𝛾𝑗subscript𝜌𝑗1𝑗1𝐽\alpha=0,\ \ \gamma_{j}=\rho_{j}=1,\ \ j=1,...,J. (1.15)

Other cases will be discussed in Section 5. The factors γjsubscript𝛾𝑗\gamma_{j} and ρjsubscript𝜌𝑗\rho_{j} in (1.13) are real positive numbers.

The variational formulation of the problem (1.6)-(1.12) reads:

a(u,w;Ξ(h))=λ(h)b(u,w;Ξ(h))wH01(Ξ(h);Γ(h))formulae-sequence𝑎𝑢𝑤Ξ𝜆𝑏𝑢𝑤Ξfor-all𝑤superscriptsubscript𝐻01ΞΓa\left(u,w;\Xi\left(h\right)\right)=\lambda(h)b\left(u,w;\Xi\left(h\right)\right)\ \ \ \forall w\in H_{0}^{1}\left(\Xi\left(h\right);\Gamma\left(h\right)\right) (1.16)

where H01(Ξ(h);Γ(h))superscriptsubscript𝐻01ΞΓH_{0}^{1}\left(\Xi\left(h\right);\Gamma\left(h\right)\right) is the Sobolev space of functions satisfying the Dirichlet conditions (1.10) on Γ(h)=ω1h(l1)ωJh(lJ),Γsuperscriptsubscript𝜔1subscript𝑙1superscriptsubscript𝜔𝐽subscript𝑙𝐽\Gamma\left(h\right)=\omega_{1}^{h}\left(l_{1}\right)\cup...\cup\omega_{J}^{h}\left(l_{J}\right),

a(u,v;Ξ(h))𝑎𝑢𝑣Ξ\displaystyle a\left(u,v;\Xi\left(h\right)\right) =(xu0,xw0)Ω(h)+jγj(h)(xuj,xwj)Ωj(h),absentsubscriptsubscript𝑥subscript𝑢0subscript𝑥subscript𝑤0subscriptΩsubscript𝑗subscript𝛾𝑗subscriptsubscript𝑥subscript𝑢𝑗subscript𝑥subscript𝑤𝑗subscriptΩ𝑗\displaystyle=\left(\nabla_{x}u_{0},\nabla_{x}w_{0}\right)_{\Omega_{\bullet}\left(h\right)}+\sum\nolimits_{j}\gamma_{j}\left(h\right)\left(\nabla_{x}u_{j},\nabla_{x}w_{j}\right)_{\Omega_{j}\left(h\right)}, (1.17)
b(u,v;Ξ(h))𝑏𝑢𝑣Ξ\displaystyle b\left(u,v;\Xi\left(h\right)\right) =(u0,w0)Ω(h)+jρj(h)(uj,wj)Ωj(h),absentsubscriptsubscript𝑢0subscript𝑤0subscriptΩsubscript𝑗subscript𝜌𝑗subscriptsubscript𝑢𝑗subscript𝑤𝑗subscriptΩ𝑗\displaystyle=\left(u_{0},w_{0}\right)_{\Omega_{\bullet}\left(h\right)}+\sum\nolimits_{j}\rho_{j}\left(h\right)\left(u_{j},w_{j}\right)_{\Omega_{j}\left(h\right)},

(,)Ξ(h)\left(\ ,\ \right)_{\Xi\left(h\right)} is the natural scalar product in the Lebesgue space L2(Ξ(h))superscript𝐿2ΞL^{2}\left(\Xi\left(h\right)\right) and jsubscript𝑗\sum\nolimits_{j} everywhere stands for summation over j=1,,J𝑗1𝐽j=1,...,J.

In view of the compact embedding H1(Ξ(h))L2(Ξ(h))superscript𝐻1Ξsuperscript𝐿2ΞH^{1}\left(\Xi\left(h\right)\right)\subset L^{2}\left(\Xi\left(h\right)\right) the spectral problem (1.16) possesses, for every fixed hh, the following positive monotone unbounded sequence of eigenvalues

0<λ1(h)<λ2(h)λn(h)+0superscript𝜆1superscript𝜆2superscript𝜆𝑛0<\lambda^{1}(h)<\lambda^{2}(h)\leq...\leq\lambda^{n}(h)\leq...\rightarrow+\infty (1.18)

listed according to their multiplicity. The corresponding eigenfunctions u1(h,),u2(h,),,un(h,),H01(Ξ(h);Γ(h))superscript𝑢1superscript𝑢2superscript𝑢𝑛superscriptsubscript𝐻01ΞΓu^{1}(h,\cdot),\ u^{2}(h,\cdot),...,\ u^{n}(h,\cdot),...\in H_{0}^{1}\left(\Xi\left(h\right);\Gamma\left(h\right)\right) can be subject to the normalization and orthogonality conditions

b(un,vm;Ξ(h))=δn,m,n,m={1,2,3,},formulae-sequence𝑏superscript𝑢𝑛superscript𝑣𝑚Ξsubscript𝛿𝑛𝑚𝑛𝑚123b\left(u^{n},v^{m};\Xi\left(h\right)\right)=\delta_{n,m},\ \ n,m\in\mathbb{N}=\left\{1,2,3,...\right\}, (1.19)

where δn,msubscript𝛿𝑛𝑚\delta_{n,m} is the Kronecker symbol.

1.3 The hybrid domain

The asymptotic analysis which has been presented at length, for example, in [4, 5, 6] and, in particular, includes the dimension reduction procedure, converts the differential equations (1.6) and (1.7) with the Neumann boundary conditions (1.8) and (1.9), respectively, into the limit equations

Δyv0(y)subscriptΔ𝑦subscript𝑣0𝑦\displaystyle-\Delta_{y}v_{0}(y) =μv0(y),yω,formulae-sequenceabsent𝜇subscript𝑣0𝑦𝑦subscript𝜔direct-product\displaystyle=\mu v_{0}(y),\ \ \ y\in\omega_{\odot}, (1.20)
γj|ωj|z2vj(z)subscript𝛾𝑗subscript𝜔𝑗superscriptsubscript𝑧2subscript𝑣𝑗𝑧\displaystyle-\gamma_{j}|\omega_{j}|\partial_{z}^{2}v_{j}(z) =μρj|ωj|vj(z),z(0,lj),formulae-sequenceabsent𝜇subscript𝜌𝑗subscript𝜔𝑗subscript𝑣𝑗𝑧𝑧0subscript𝑙𝑗\displaystyle=\mu\rho_{j}|\omega_{j}|v_{j}(z),\ \ \ z\in(0,l_{j}), (1.21)

where ωsubscript𝜔direct-product\omega_{\odot} is the punctured domain ω0𝒫,subscript𝜔0𝒫\omega_{0}\setminus\mathcal{P}, 𝒫={P1,,PJ}𝒫superscript𝑃1superscript𝑃𝐽\mathcal{P}=\left\{P^{1},...,P^{J}\right\} and |ωj|subscript𝜔𝑗|\omega_{j}| is the area of the domain ωjsubscript𝜔𝑗\omega_{j}. Moreover, a primary examination of the boundary layer phenomenon near the lateral side υ0(h)subscript𝜐0\mathbb{\upsilon}_{0}\left(h\right) of the plate and the end ωjh(lj)superscriptsubscript𝜔𝑗subscript𝑙𝑗\omega_{j}^{h}(l_{j}) of the rod Ωj(h)subscriptΩ𝑗\Omega_{j}\left(h\right), respectively, gives the following boundary conditions

νv0(y)subscript𝜈subscript𝑣0𝑦\displaystyle\partial_{\nu}v_{0}(y) =0,yω0,formulae-sequenceabsent0𝑦subscript𝜔0\displaystyle=0,\ \ \ y\in\partial\omega_{0}, (1.22)
vj(lj)subscript𝑣𝑗subscript𝑙𝑗\displaystyle v_{j}(l_{j}) =0.absent0\displaystyle=0. (1.23)

However, the one-dimensional and two-dimensional problems are not completed yet due to the lack of boundary conditions at the endpoints z=0𝑧0z=0 of the intervals (0,lj)0subscript𝑙𝑗(0,l_{j}) and because the differential equation (1.20) is fulfilled for sure only outside the points P1,,PJsuperscript𝑃1superscript𝑃𝐽P^{1},...,P^{J}, since near the sockets ωjh×(0,h)superscriptsubscript𝜔𝑗0\omega_{j}^{h}\times(0,h) the geometrical structure of the junction (1.4) changes and becomes crucially spacial so that the dimension reduction does not work. As was shown in [4], this observation requires to consider solutions of the problem (1.20), (1.22) with logarithmic singularities at the points in the set 𝒫𝒫\mathcal{P}. We also will take in the sequel such singular solutions into account, however further considerations in this paper diverge from the asymptotic analysis used in [4]. Indeed, we will provide two abstract but applicable formulations of a spectral problem in the hybrid domain in fig. 2,a, which give an approximation of the spectrum (1.18) with relatively high precision. First, we detect a self-adjoint operator as an extension of the differential operator of the problem (1.20)-(1.23) supplied (cf. (2.3) and (2.1)) with the restrictive conditions

v0(Pj)=0,vj(0)=zvj(0)=0,j=1,,J.formulae-sequenceformulae-sequencesubscript𝑣0superscript𝑃𝑗0subscript𝑣𝑗0subscript𝑧subscript𝑣𝑗00𝑗1𝐽v_{0}(P^{j})=0,\ \ v_{j}(0)=\partial_{z}v_{j}(0)=0,\ \ j=1,...,J. (1.24)

Second, we construct certain point conditions at P1,,PJsuperscript𝑃1superscript𝑃𝐽P^{1},...,P^{J} which tie the independent problems (1.20), (1.22) and (1.21), (1.23) into a formally self-adjoint problem in the hybrid domain Ξ0.superscriptΞ0\Xi^{0}. These two formulations happen to be equivalent and both are realized as operators with the discrete spectrum which, in the low-frequency range, approximate the spectrum of the problem (1.6)-(1.12) (or, equivalently, (1.16)) with admissible precision111The error estimates are derived in the paper with quite simple tools. Advanced estimation may detect the accuracy O(h|lnh|)𝑂O(h|\ln h|) and extend the proximity property of the models to a part of the mid-frequency range, cf. [20]. The latter, however, enlarges enormously massif of calculations. O(h1/2|lnh|3)𝑂superscript12superscript3O(h^{1/2}|\ln h|^{3}). Unfortunately, serving for a particular range of the spectrum, both the operators lose the positivity property and gain so called parasite eigenvalues which are negative and big, of order h2superscript2h^{-2}; therefore, we prove that they lay outside the scope of the asymptotic models and have no relation to the original problem. In order to furnish, for example, an application of the minimum principle, cf. [2, Thm. 10.2.1], we construct detailed asymptotics of these parasite eigenvalues and of the corresponding eigenfunctions, which are located in the very vicinity of the points P1,,PJsuperscript𝑃1superscript𝑃𝐽P^{1},...,P^{J} and decay exponentially at a distance from them.

Parameters of the self-adjoint extension and ingredients of the point conditions are found out with the help of the method of matched asymptotic expansions on the basis of special solutions described in the first part [4] of our work. Both linearly depend on |lnh||\ln h| and this makes the eigenvalues and eigenvectors to be real analytic functions in |lnh|1.superscript1|\ln h|^{-1}. However, a possible numerical realization of the models with a small but fixed parameter hh does not require to take into account such complication of asymptotic expansions. We, of course, compute explicitly couple of initial terms of the convergent series in |lnh|1.superscript1|\ln h|^{-1}.

1.4 Outline of the paper

In Section 2 we describe all self-adjoint extensions of the operator of the problem (1.20)-(1.24) as well as the point conditions which involve the problems (1.20), (1.22) and (1.21), (1.23) into a symmetric generalized Green formula. This material is known and is presented in a condensed form, mainly in order to introduce the notation and explain some technicalities used throughout the paper. We refer to the review papers [29], [3] and [21] for a detailed information. If the skeleton Ξ(0)Ξ0\Xi(0) decouples in the limit, see fig. 2,b, then the extended operator of the Neumann problem (1.20), (1.22) may require for ”potentials of zero radii” [1, 29] or ”pseudo-Laplacian” in the terminology [7], but in this case the ordinary differential equations (1.21), (1.23) are supplied with either Neumann, or Dirichlet condition at the endpoints z=0𝑧0z=0 of the interval (cf., Remark 5, 6 and see the paper [3] which provides the complete description of the techniques of self-adjoint extensions).

The most interesting situation occurs under the restriction (1.14) when the skeleton does not decouple in the limit, see fig. 2,a. The asymptotic procedure in [4] allow us to determine in Section 3 appropriate parameters of a particular self-adjoint extension serving for the original problem (1.6)-(1.12) as well as all ingredients of the point condition in the corresponding hybrid model.

The most cumbersome part of our analysis is concentrated in Section 4, where we perform the justification of our asymptotic models with the help of weighted estimates obtained in [4]. We state here the main result of the paper, Theorem 16. Section 5 contains some simple asymptotic formulas and the example of the homogeneous junction, see (1.15).

2 General statement of problems on the hybrid domain

2.1 Unbounded operators and their adjoints

Let Ajsubscript𝐴𝑗A_{j} be an unbounded operator in the Lebesgue space L2(Ij)superscript𝐿2subscript𝐼𝑗L^{2}(I_{j}) with the differential expression γj|ωj|z2subscript𝛾𝑗subscript𝜔𝑗superscriptsubscript𝑧2-\gamma_{j}|\omega_{j}|\partial_{z}^{2} and the domain

𝒟(Aj)={wjH2(Ij):vj(lj)=0,vj(0)=zvj(0)=0}.𝒟subscript𝐴𝑗conditional-setsubscript𝑤𝑗superscript𝐻2subscript𝐼𝑗formulae-sequencesubscript𝑣𝑗subscript𝑙𝑗0subscript𝑣𝑗0subscript𝑧subscript𝑣𝑗00\mathcal{D}(A_{j})=\left\{w_{j}\in H^{2}(I_{j}):v_{j}(l_{j})=0,\ v_{j}(0)=\partial_{z}v_{j}(0)=0\right\}. (2.1)

The operator is symmetric and closed. By a direct calculation, it follows that the adjoint operator Ajsuperscriptsubscript𝐴𝑗A_{j}^{\ast} gets the same differential expression but its domain is bigger, namely

𝒟(Aj)={wjH2(Ij):vj(lj)=0}.𝒟superscriptsubscript𝐴𝑗conditional-setsubscript𝑤𝑗superscript𝐻2subscript𝐼𝑗subscript𝑣𝑗subscript𝑙𝑗0\mathcal{D}(A_{j}^{\ast})=\left\{w_{j}\in H^{2}(I_{j}):v_{j}(l_{j})=0\right\}. (2.2)

Hence, dim(𝒟(Aj)/𝒟(Aj))=2dimension𝒟superscriptsubscript𝐴𝑗𝒟subscript𝐴𝑗2\dim(\mathcal{D}(A_{j}^{\ast})/\mathcal{D}(A_{j}))=2 and the defect index of Ajsubscript𝐴𝑗A_{j} is 1:1:111:1.

Analogously, we introduce the unbounded operator A0subscript𝐴0A_{0} in L2(ω0)superscript𝐿2subscript𝜔0L^{2}(\omega_{0}) with the differential expression Δy=2/y122/y22subscriptΔ𝑦superscript2superscriptsubscript𝑦12superscript2superscriptsubscript𝑦22-\Delta_{y}=-\partial^{2}/\partial y_{1}^{2}-\partial^{2}/\partial y_{2}^{2} and the domain

𝒟(A0)={w0H2(ω0):νw0=0 on ω0,wj(Pk)=0,k=1,,J}.𝒟subscript𝐴0conditional-setsubscript𝑤0superscript𝐻2subscript𝜔0formulae-sequencesubscript𝜈subscript𝑤00 on subscript𝜔0formulae-sequencesubscript𝑤𝑗superscript𝑃𝑘0𝑘1𝐽\mathcal{D}(A_{0})=\left\{w_{0}\in H^{2}(\omega_{0}):\partial_{\nu}w_{0}=0\text{ on }\partial\omega_{0},\ w_{j}(P^{k})=0,\ k=1,...,J\right\}. (2.3)

By virtue of the Sobolev embedding theorem H2(ω0)C(ω0)superscript𝐻2subscript𝜔0𝐶subscript𝜔0H^{2}(\omega_{0})\subset C(\omega_{0}) and the classical Green formula

(Δyw0,v0)ω0+(νw0,v0)ω0=(w0,Δyv0)ω0+(w0,νv0)ω0,subscriptsubscriptΔ𝑦subscript𝑤0subscript𝑣0subscript𝜔0subscriptsubscript𝜈subscript𝑤0subscript𝑣0subscript𝜔0subscriptsubscript𝑤0subscriptΔ𝑦subscript𝑣0subscript𝜔0subscriptsubscript𝑤0subscript𝜈subscript𝑣0subscript𝜔0-(\Delta_{y}w_{0},v_{0})_{\omega_{0}}+(\partial_{\nu}w_{0},v_{0})_{\partial\omega_{0}}=-(w_{0},\Delta_{y}v_{0})_{\omega_{0}}+(w_{0},\partial_{\nu}v_{0})_{\partial\omega_{0}}, (2.4)

the operator A0subscript𝐴0A_{0} is closed and symmetric. The following lemma, in particular, shows that the defect index of A0subscript𝐴0A_{0} is J×J𝐽𝐽J\times J.

Lemma 1

The adjoint operator A0superscriptsubscript𝐴0A_{0}^{\ast} for A0subscript𝐴0A_{0} has the differential expression ΔysubscriptΔ𝑦-\Delta_{y} and the domain

𝒟(A0)𝒟superscriptsubscript𝐴0\displaystyle\mathcal{D}(A_{0}^{\ast}) ={V0L2(ω0):V0(y)=V^0(y)12πjbjχj(y)ln|yPj|,\displaystyle=\left\{V_{0}\in L^{2}(\omega_{0}):V_{0}(y)=\widehat{V}_{0}(y)-\frac{1}{2\pi}{\textstyle\sum\nolimits_{j}}b_{j}\chi_{j}(y)\ln|y-P^{j}|,\right.\ (2.5)
bj,V^0(y)H2(ω0),νV^0=0 on ω0},\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left.b_{j}\in\mathbb{C},\ \widehat{V}_{0}(y)\in H^{2}(\omega_{0}),\ \partial_{\nu}\widehat{V}_{0}=0\text{ on }\partial\omega_{0}\right\},

where χ1,,χJCc(ω0)subscript𝜒1subscript𝜒𝐽superscriptsubscript𝐶𝑐subscript𝜔0\chi_{1},...,\chi_{J}\in C_{c}^{\infty}(\omega_{0}) are cut-off functions such that

χj(Pj)=1,χj(y)χk(y)=0 for jk,suppχjω0.formulae-sequenceformulae-sequencesubscript𝜒𝑗superscript𝑃𝑗1subscript𝜒𝑗𝑦subscript𝜒𝑘𝑦0 for 𝑗𝑘suppsubscript𝜒𝑗subscript𝜔0\chi_{j}(P^{j})=1,\ \ \ \chi_{j}(y)\chi_{k}(y)=0\text{ for }j\neq k,\ \ \ \mathrm{supp}\chi_{j}\subset\omega_{0}.

Proof. By definition, a function V0L2(ω0)subscript𝑉0superscript𝐿2subscript𝜔0V_{0}\in L^{2}(\omega_{0}) belongs to 𝒟(A0)𝒟superscriptsubscript𝐴0\mathcal{D}(A_{0}^{\ast}) if and only if the following integral identity holds:

(V0,Δyv0)ω0=(F0,v0)ω0v0𝒟(A0).formulae-sequencesubscriptsubscript𝑉0subscriptΔ𝑦subscript𝑣0subscript𝜔0subscriptsubscript𝐹0subscript𝑣0subscript𝜔0for-allsubscript𝑣0𝒟subscript𝐴0-(V_{0},\Delta_{y}v_{0})_{\omega_{0}}=(F_{0},v_{0})_{\omega_{0}}\ \ \ \forall v_{0}\in\mathcal{D}(A_{0}). (2.6)

At the first step we take v0Cc(ω¯0𝒫)𝒟(A0).subscript𝑣0superscriptsubscript𝐶𝑐subscript¯𝜔0𝒫𝒟subscript𝐴0v_{0}\in C_{c}^{\infty}(\overline{\omega}_{0}\setminus\mathcal{P})\cap\mathcal{D}(A_{0}). Based on the Green formula (2.4), we recall the Neumann boundary condition in (2.3) and apply classical results [15, §3-6, Ch. 2] on lifting smoothness of solutions to elliptic problems. In this way we conclude that V0Hloc2(ω¯0𝒫)subscript𝑉0superscriptsubscript𝐻𝑙𝑜𝑐2subscript¯𝜔0𝒫V_{0}\in H_{loc}^{2}(\overline{\omega}_{0}\setminus\mathcal{P}) and

ΔyV0(y)=F0(y),yω=ω0𝒫,νV0(y)=0yω0.formulae-sequenceformulae-sequencesubscriptΔ𝑦subscript𝑉0𝑦subscript𝐹0𝑦𝑦subscript𝜔direct-productsubscript𝜔0𝒫subscript𝜈subscript𝑉0𝑦0𝑦subscript𝜔0-\Delta_{y}V_{0}(y)=F_{0}(y),\ y\in\omega_{\odot}=\omega_{0}\setminus\mathcal{P},\ \ \ \ \partial_{\nu}V_{0}(y)=0\text{, }y\in\partial\omega_{0}. (2.7)

The next step requires for results [8] of the theory of elliptic problem in domains with conical points. Indeed, regarding Pjsuperscript𝑃𝑗P^{j} as the top of the ”complete cone” 2Pjsuperscript2superscript𝑃𝑗\mathbb{R}^{2}\setminus P^{j}, that is, the punctured plane, we introduce the Kondratiev space Vβl(ω0)superscriptsubscript𝑉𝛽𝑙subscript𝜔0V_{\beta}^{l}(\omega_{0}) with l{0,1,2,}𝑙012l\in\left\{0,1,2,...\right\} and β𝛽\beta\in\mathbb{R} as the completion of Cc(ω¯0𝒫)superscriptsubscript𝐶𝑐subscript¯𝜔0𝒫C_{c}^{\infty}(\overline{\omega}_{0}\setminus\mathcal{P}) with respect to the weighted norm

||v0;Vβl(ω0)||=(k=0l||min{r1,,rJ}βlkykv0;L2(ω0)2||)1/2||v_{0};V_{\beta}^{l}(\omega_{0})||=\left({\textstyle\sum\limits_{k=0}^{l}}||\min\left\{r_{1},...,r_{J}\right\}^{\beta-l-k}\nabla_{y}^{k}v_{0};L^{2}(\omega_{0})^{2}||\right)^{1/2} (2.8)

where rj=|yPj|subscript𝑟𝑗𝑦superscript𝑃𝑗r_{j}=|y-P^{j}| and ykv0superscriptsubscript𝑦𝑘subscript𝑣0\nabla_{y}^{k}v_{0} stands for a collection of all order k𝑘k derivatives of v0subscript𝑣0v_{0}. Clearly, L2(ω0)Vδ0(ω0)superscript𝐿2subscript𝜔0superscriptsubscript𝑉𝛿0subscript𝜔0L^{2}(\omega_{0})\subset V_{\delta}^{0}(\omega_{0}) and Vδ2(ω0)H1(ω0)superscriptsubscript𝑉𝛿2subscript𝜔0superscript𝐻1subscript𝜔0V_{\delta}^{2}(\omega_{0})\subset H^{1}(\omega_{0}) for δ[0,1]𝛿01\delta\in[0,1].

Since V0L2(ω0)V00(ω0)subscript𝑉0superscript𝐿2subscript𝜔0superscriptsubscript𝑉00subscript𝜔0V_{0}\in L^{2}(\omega_{0})\subset V_{0}^{0}(\omega_{0}), the theorem on asymptotics [8], see, e.g., [25, §3.5, §4.2, §6.4] and also the introductory chapters in the books [25, 10], gives the representation

V0(y)=V~0(y)+jχj(y)(ajbj2πlnrj)subscript𝑉0𝑦subscript~𝑉0𝑦subscript𝑗subscript𝜒𝑗𝑦subscript𝑎𝑗subscript𝑏𝑗2𝜋subscript𝑟𝑗V_{0}(y)=\widetilde{V}_{0}(y)+{\textstyle\sum\nolimits_{j}}\chi_{j}(y)\left(a_{j}-\frac{b_{j}}{2\pi}\ln r_{j}\right) (2.9)

as well as the inclusion V~0Vδ2(ω0)subscript~𝑉0superscriptsubscript𝑉𝛿2subscript𝜔0\widetilde{V}_{0}\in V_{\delta}^{2}(\omega_{0}) with any δ>0𝛿0\delta>0 and the estimate

||V~0;Vδ2(ω0)||+j(|aj|+|bj|)cδ(F0;L2(ω0)+V0;L2(ω0)).||\widetilde{V}_{0};V_{\delta}^{2}(\omega_{0})||+{\textstyle\sum\nolimits_{j}}(|a_{j}|+|b_{j}|)\leq c_{\delta}(\left\|F_{0};L^{2}(\omega_{0})\right\|+\left\|V_{0};L^{2}(\omega_{0})\right\|). (2.10)

Hence, the sum

V^0(y)=V0(y)+12πjbjlnrjsubscript^𝑉0𝑦subscript𝑉0𝑦12𝜋subscript𝑗subscript𝑏𝑗subscript𝑟𝑗\widehat{V}_{0}(y)=V_{0}(y)+\frac{1}{2\pi}\sum\nolimits_{j}b_{j}\ln r_{j} (2.11)

belongs to H1(ω0)superscript𝐻1subscript𝜔0H^{1}(\omega_{0}) and still solves the problem (2.7) with a new right-hand side F^0L2(ω0)subscript^𝐹0superscript𝐿2subscript𝜔0\widehat{F}_{0}\in L^{2}(\omega_{0}) in the Poisson equation. Thus, referring to [15, §9, Ch. 2] we have V^0H2(ω0)subscript^𝑉0superscript𝐻2subscript𝜔0\widehat{V}_{0}\in H^{2}(\omega_{0}) and, therefore, V0subscript𝑉0V_{0} falls into the linear set (2.5). \blacksquare

Remark 2

The representation (2.9) can be derived by means of the Fourier method but the Kondratiev theory [8] helps to avoid any calculation.

2.2 The generalized Green formula

The norm, cf. the left-hand side of (2.10),

V0;0=(||V^0;H2(ω0)||2+j|bj|2)1/2\left\|V_{0};\mathfrak{H}_{0}\right\|=(||\widehat{V}_{0};H^{2}(\omega_{0})||^{2}+{\textstyle\sum\nolimits_{j}}|b_{j}|^{2})^{1/2} (2.12)

brings Hilbert structure to the linear space (2.5) denoted by 0subscript0\mathfrak{H}_{0}. By \mathfrak{H}, we understand the direct product

=0×1××Jsubscript0subscript1subscript𝐽\mathfrak{H}=\mathfrak{H}_{0}\times\mathfrak{H}_{1}\times...\times\mathfrak{H}_{J} (2.13)

where jsubscript𝑗\mathfrak{H}_{j} is the linear space (2.2) with the Sobolev H2superscript𝐻2H^{2}-norm. Moreover, taking into account the right-hand sides of the equations (1.20) and (1.21) we supply the vector Lebesgue space 𝔏𝔏\mathfrak{L} with the special norm

||v;𝔏||=(||v0;L2(ω0)||2+jρj|ωj|||vj;L2(Ij)||2)1/2,||v;\mathfrak{L}||=(||v_{0};L^{2}(\omega_{0})||^{2}+{\textstyle\sum\nolimits_{j}}\rho_{j}|\omega_{j}|\ ||v_{j};L^{2}(I_{j})||^{2})^{1/2}, (2.14)

with v=(v0,v1,,vJ)𝔏:=L2(ω0)×L2(I1)××L2(IJ).𝑣subscript𝑣0subscript𝑣1subscript𝑣𝐽𝔏assignsuperscript𝐿2subscript𝜔0superscript𝐿2subscript𝐼1superscript𝐿2subscript𝐼𝐽v=(v_{0},v_{1},...,v_{J})\in\mathfrak{L}:=L^{2}(\omega_{0})\times L^{2}(I_{1})\times...\times L^{2}(I_{J}).

We also introduce two continuous projections ±:2J:subscriptWeierstrass-pplus-or-minussuperscript2𝐽\wp_{\pm}:\mathfrak{H}\rightarrow\mathbb{R}^{2J} by the formulas

+vsubscriptWeierstrass-p𝑣\displaystyle\wp_{+}v =(+v,+′′v)=(v^0(P1),,v^0(PJ),v1(0),,vJ(0)),absentsuperscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p′′𝑣subscript^𝑣0superscript𝑃1subscript^𝑣0superscript𝑃𝐽subscript𝑣10subscript𝑣𝐽0\displaystyle=(\wp_{+}^{\prime}v,\wp_{+}^{\prime\prime}v)=(\widehat{v}_{0}(P^{1}),...,\widehat{v}_{0}(P^{J}),v_{1}(0),...,v_{J}(0)), (2.15)
vsubscriptWeierstrass-p𝑣\displaystyle\wp_{-}v =(v,′′v)=(b1,,bJ,γ1|ω1|zv1(0),,γJ|ωJ|zvJ(0)),absentsuperscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p′′𝑣subscript𝑏1subscript𝑏𝐽subscript𝛾1subscript𝜔1subscript𝑧subscript𝑣10subscript𝛾𝐽subscript𝜔𝐽subscript𝑧subscript𝑣𝐽0\displaystyle=(\wp_{-}^{\prime}v,\wp_{-}^{\prime\prime}v)=(b_{1},...,b_{J},-\gamma_{1}|\omega_{1}|\partial_{z}v_{1}(0),...,-\gamma_{J}|\omega_{J}|\partial_{z}v_{J}(0)),

where v=(v0,v1,,vJ)𝑣subscript𝑣0subscript𝑣1subscript𝑣𝐽v=(v_{0},v_{1},...,v_{J})\in\mathfrak{H} and b1,,bJsubscript𝑏1subscript𝑏𝐽b_{1},...,b_{J}, v^0subscript^𝑣0\widehat{v}_{0} are attributes of the decomposition (2.9) of v00subscript𝑣0subscript0v_{0}\in\mathfrak{H}_{0}.

The next assertion is but a concretization of a general result in [27, 24, 26], see also [25, §6.2], however we give a condensed and much simplified proof for reader’s convenience.

Proposition 3

For v𝑣v and w=(w0,w1,,wJ)𝑤subscript𝑤0subscript𝑤1subscript𝑤𝐽w=(w_{0},w_{1},...,w_{J}) in \mathfrak{H}, the generalized Green formula

q(v,w)𝑞𝑣𝑤\displaystyle q(v,w) :=(Δyv0,w0)ω0+(v0,Δyw0)ω0jγj|ωj|((z2vj,wj)Ij(vj,z2wj)Ij)assignabsentsubscriptsubscriptΔ𝑦subscript𝑣0subscript𝑤0subscript𝜔0subscriptsubscript𝑣0subscriptΔ𝑦subscript𝑤0subscript𝜔0subscript𝑗subscript𝛾𝑗subscript𝜔𝑗subscriptsuperscriptsubscript𝑧2subscript𝑣𝑗subscript𝑤𝑗subscript𝐼𝑗subscriptsubscript𝑣𝑗superscriptsubscript𝑧2subscript𝑤𝑗subscript𝐼𝑗\displaystyle:=-(\Delta_{y}v_{0},w_{0})_{\omega_{0}}+(v_{0},\Delta_{y}w_{0})_{\omega_{0}}-{\textstyle\sum\nolimits_{j}}\gamma_{j}|\omega_{j}|((\partial_{z}^{2}v_{j},w_{j})_{I_{j}}-(v_{j},\partial_{z}^{2}w_{j})_{I_{j}}) (2.16)
=+v,wv,+wabsentsubscriptWeierstrass-p𝑣subscriptWeierstrass-p𝑤subscriptWeierstrass-p𝑣subscriptWeierstrass-p𝑤\displaystyle=\left\langle\wp_{+}v,\wp_{-}w\right\rangle-\left\langle\wp_{-}v,\wp_{+}w\right\rangle

is valid, where ,\left\langle\ ,\ \right\rangle stands for the natural scalar product in 2Jsuperscript2𝐽\mathbb{R}^{2J}.

Proof. First of all, we write the evident identity

(z2vj,wj)Ij+(vj,z2wj)Ij=wj(0)zvj(0)vj(0)zwj(0)subscriptsuperscriptsubscript𝑧2subscript𝑣𝑗subscript𝑤𝑗subscript𝐼𝑗subscriptsubscript𝑣𝑗superscriptsubscript𝑧2subscript𝑤𝑗subscript𝐼𝑗subscript𝑤𝑗0subscript𝑧subscript𝑣𝑗0subscript𝑣𝑗0subscript𝑧subscript𝑤𝑗0-(\partial_{z}^{2}v_{j},w_{j})_{I_{j}}+(v_{j},\partial_{z}^{2}w_{j})_{I_{j}}=w_{j}(0)\partial_{z}v_{j}(0)-v_{j}(0)\partial_{z}w_{j}(0) (2.17)

and multiply it with γj|ωj|subscript𝛾𝑗subscript𝜔𝑗\gamma_{j}|\omega_{j}|. Then we take v0,w00subscript𝑣0subscript𝑤0subscript0v_{0},w_{0}\in\mathfrak{H}_{0} and write the standard Green formula

(\displaystyle-( Δyv0,w0)ω0+(v0,Δyw0)ω0=limδ+0((Δyv0,w0)ωδ+(v0,Δyw0)ωδ)\displaystyle\Delta_{y}v_{0},w_{0})_{\omega_{0}}+(v_{0},\Delta_{y}w_{0})_{\omega_{0}}=\lim_{\delta\rightarrow+0}(-(\Delta_{y}v_{0},w_{0})_{\omega_{\delta}}+(v_{0},\Delta_{y}w_{0})_{\omega_{\delta}}) (2.18)
=limδ+0jδ02π((w^(Pj)bjw2πlnrj)rjbjv2πlnrj(v^(Pj)bjv2πlnrj)rjbjw2πlnrj)𝑑φjabsentsubscript𝛿0subscript𝑗𝛿superscriptsubscript02𝜋^𝑤superscript𝑃𝑗superscriptsubscript𝑏𝑗𝑤2𝜋subscript𝑟𝑗subscript𝑟𝑗superscriptsubscript𝑏𝑗𝑣2𝜋subscript𝑟𝑗^𝑣superscript𝑃𝑗superscriptsubscript𝑏𝑗𝑣2𝜋subscript𝑟𝑗subscript𝑟𝑗superscriptsubscript𝑏𝑗𝑤2𝜋subscript𝑟𝑗differential-dsubscript𝜑𝑗\displaystyle=-\lim_{\delta\rightarrow+0}\sum\nolimits_{j}\delta\int_{0}^{2\pi}\left(\left(\widehat{w}(P^{j})-\frac{b_{j}^{w}}{2\pi}\ln r_{j}\right)\frac{\partial}{\partial r_{j}}\frac{b_{j}^{v}}{2\pi}\ln r_{j}-\left(\widehat{v}(P^{j})-\frac{b_{j}^{v}}{2\pi}\ln r_{j}\right)\frac{\partial}{\partial r_{j}}\frac{b_{j}^{w}}{2\pi}\ln r_{j}\right)d\varphi_{j}
=j(w^(Pj)bjvv^(Pj)bjw)absentsubscript𝑗^𝑤superscript𝑃𝑗superscriptsubscript𝑏𝑗𝑣^𝑣superscript𝑃𝑗superscriptsubscript𝑏𝑗𝑤\displaystyle=-\sum\nolimits_{j}(\widehat{w}(P^{j})b_{j}^{v}-\widehat{v}(P^{j})b_{j}^{w})

where ωδ=ω0𝔹δ(Pj),subscript𝜔𝛿subscript𝜔0subscript𝔹𝛿superscript𝑃𝑗\omega_{\delta}=\omega_{0}\setminus\mathbb{B}_{\delta}(P^{j}), 𝔹δ(Pj)={y:rj<δ}subscript𝔹𝛿superscript𝑃𝑗conditional-set𝑦subscript𝑟𝑗𝛿\mathbb{B}_{\delta}(P^{j})=\{y:r_{j}<\delta\} is a disk and (rj,φj)+×[0,2π)subscript𝑟𝑗subscript𝜑𝑗subscript02𝜋(r_{j},\varphi_{j})\in\mathbb{R}_{+}\times[0,2\pi) is the polar coordinate system centered at Pjsuperscript𝑃𝑗P^{j}. Now (2.16) follows from (2.17), (2.18) and (2.15). \blacksquare

2.3 Self-adjoint extensions

Calculations in Section 2.2 detect the defect index 2J:2J:2𝐽2𝐽2J:2J of the operator A=(A0,A1,,AJ)𝐴subscript𝐴0subscript𝐴1subscript𝐴𝐽A=(A_{0},A_{1},...,A_{J}) with the differential expression (Δyv0,γ1ρ11z2,,γJρJ1z2)subscriptΔ𝑦subscript𝑣0subscript𝛾1superscriptsubscript𝜌11superscriptsubscript𝑧2subscript𝛾𝐽superscriptsubscript𝜌𝐽1superscriptsubscript𝑧2(-\Delta_{y}v_{0},-\gamma_{1}\rho_{1}^{-1}\partial_{z}^{2},...,-\gamma_{J}\rho_{J}^{-1}\partial_{z}^{2}) in the Hilbert space 𝔏𝔏\mathfrak{L}, see (2.14). Hence, this operator admits a self-adjoint extension 𝒜𝒜\mathcal{A}, that is, A𝒜A𝐴𝒜superscript𝐴A\subset\mathcal{A}\subset A^{\ast} and 𝒜=𝒜𝒜superscript𝒜\mathcal{A}=\mathcal{A}^{\ast}.

Since 𝒜𝒜\mathcal{A} is a restriction of A=(A0,A1,,AJ)superscript𝐴superscriptsubscript𝐴0superscriptsubscript𝐴1superscriptsubscript𝐴𝐽A^{\ast}=(A_{0}^{\ast},A_{1}^{\ast},...,A_{J}^{\ast}), we conclude that

𝒟(𝒜)=𝒟(𝒜)={v𝒟(𝒜):(+v,v)}𝒟𝒜𝒟superscript𝒜conditional-set𝑣𝒟𝒜subscriptWeierstrass-p𝑣subscriptWeierstrass-p𝑣\mathcal{D}(\mathcal{A})=\mathcal{D}(\mathcal{A}^{\ast})=\left\{v\in\mathcal{D}(\mathcal{A}):(\wp_{+}v,\wp_{-}v)\in\mathcal{R}\right\} (2.19)

where a linear subspace 4Jsuperscript4𝐽\mathcal{R}\subset\mathbb{R}^{4J} of dimension 2J2𝐽2J must be chosen such that in accord with Proposition 3

q(v,w)=0v,w𝒟(𝒜).formulae-sequence𝑞𝑣𝑤0for-all𝑣𝑤𝒟𝒜q(v,w)=0\ \ \ \forall v,w\in\mathcal{D}(\mathcal{A}). (2.20)

The symplectic form (2.16) is actually defined on the factor space 𝒟(𝒜)/𝒟(𝒜)4J𝒟superscript𝒜𝒟𝒜superscript4𝐽\mathcal{D}(\mathcal{A}^{\ast})/\mathcal{D}(\mathcal{A})\approx\mathbb{R}^{4J} because the generalized Green formula demonstrates that

0=q(v,w)=q(w,v)¯ for v𝒟(𝒜),w𝒟(𝒜).formulae-sequence0𝑞𝑣𝑤¯𝑞𝑤𝑣 for 𝑣𝒟𝒜𝑤𝒟superscript𝒜0=q(v,w)=-\overline{q(w,v)}\text{ \ for }v\in\mathcal{D}(\mathcal{A}),\ w\in\mathcal{D}(\mathcal{A}^{\ast}). (2.21)

Description of null spaces of a symplectic form in Euclidean space is a primary algebraic question, cf. [13], but it gives a direct identification of all self-adjoint extensions of our operator A𝐴A, see [1], [31], [7] and, e.g., [29, 3, 22].

Proposition 4

Let +0direct-sumsuperscriptsuperscript0superscript\mathcal{R}^{+}\mathcal{\oplus R}^{0}\mathcal{\oplus R}^{-} be an orthogonal decomposition of 2Jsuperscript2𝐽\mathbb{R}^{2J} and let 𝒮𝒮\mathcal{S} be a symmetric invertible operator in 0superscript0\mathcal{R}^{0}. The restriction 𝒜𝒜\mathcal{A} of the operator Asuperscript𝐴A^{\ast} onto the domain

𝒟(𝒜)={v𝒟(A):+v=t++𝒯t0,v=t+t0,tαα,α=0,±}𝒟𝒜conditional-set𝑣𝒟superscript𝐴formulae-sequencesubscriptWeierstrass-p𝑣superscript𝑡𝒯superscript𝑡0formulae-sequencesubscriptWeierstrass-p𝑣superscript𝑡superscript𝑡0formulae-sequencesuperscript𝑡𝛼superscript𝛼𝛼0plus-or-minus\mathcal{D}(\mathcal{A})=\left\{v\in\mathcal{D}(A^{\ast}):\wp_{+}v=t^{+}+\mathcal{T}t^{0},\ \wp_{-}v=t^{-}+t^{0},\ t^{\alpha}\in\mathcal{R}^{\alpha},\ \alpha=0,\pm\right\} (2.22)

is a self-adjoint extension of the operator A𝐴A in 𝔏𝔏\mathfrak{L}. Any self-adjoint extension of A𝐴A can be obtained in this way.

Remark 5

If we put

=0={0}2J,+=2J,formulae-sequencesuperscriptsuperscript0superscript02𝐽superscriptsuperscript2𝐽\mathcal{R}^{-}=\mathcal{R}^{0}=\{0\}^{2J},\ \ \mathcal{R}^{+}=\mathbb{R}^{2J}, (2.23)

then the self-adjoint extension 𝒜0superscript𝒜0\mathcal{A}^{0} given in Proposition 4 is nothing but the set of the two-dimensional Neumann problem

Δyv0(y)=f0(y),yω0,νv0(y)=0,yω0,formulae-sequencesubscriptΔ𝑦subscript𝑣0𝑦subscript𝑓0𝑦formulae-sequence𝑦subscript𝜔0formulae-sequencesubscript𝜈subscript𝑣0𝑦0𝑦subscript𝜔0-\Delta_{y}v_{0}(y)=f_{0}(y),\ \ y\in\omega_{0},\ \ \ \ \partial_{\nu}v_{0}(y)=0,\ \ y\in\partial\omega_{0}, (2.24)

and the one-dimensional mixed boundary-value problems

γj|ωj|z2vj(z)=fj(z),z(0,lj),vj(lj)=γj|ωj|zvj(0)=0.formulae-sequencesubscript𝛾𝑗subscript𝜔𝑗superscriptsubscript𝑧2subscript𝑣𝑗𝑧subscript𝑓𝑗𝑧formulae-sequence𝑧0subscript𝑙𝑗subscript𝑣𝑗subscript𝑙𝑗subscript𝛾𝑗subscript𝜔𝑗subscript𝑧subscript𝑣𝑗00-\gamma_{j}|\omega_{j}|\partial_{z}^{2}v_{j}(z)=f_{j}(z),\ z\in(0,l_{j}),\ \ \ v_{j}(l_{j})=\gamma_{j}|\omega_{j}|\partial_{z}v_{j}(0)=0. (2.25)

These problems are independent and are posed on the spaces H2(ω0)superscript𝐻2subscript𝜔0H^{2}(\omega_{0}) and H2(Ij)superscript𝐻2subscript𝐼𝑗H^{2}(I_{j}), respectively. The corresponding operator has the kernel spanned over the constant vectors (c0,0,,0)subscript𝑐000(c_{0},0,...,0). \blacksquare

Remark 6

In the case ={0}J×J,superscriptsuperscript0𝐽superscript𝐽\mathcal{R}^{-}=\{0\}^{J}\times\mathbb{R}^{J}, 0={0}2J,+=J×{0},formulae-sequencesuperscript0superscript02𝐽superscriptsuperscript𝐽0\mathcal{R}^{0}=\{0\}^{2J},\ \mathcal{R}^{+}=\mathbb{R}^{J}\times\{0\}, we obtain a self-adjoint extension which gives rise to the Neumann problem (2.24) and a set of the Dirichlet problems

γj|ωj|z2vj(z)=fj(z),z(0,lj),vj(lj)=vj(0)=0.-\gamma_{j}|\omega_{j}|\partial_{z}^{2}v_{j}(z)=f_{j}(z),\ z\in(0,l_{j}),\ \ \ v_{j}(l_{j})=v_{j}(0)=0.\ \ \blacksquare

In Section 3 we come across a self-adjoint extension 𝒜=𝒜h𝒜superscript𝒜\mathcal{A}=\mathcal{A}^{h} (the superscript will appear in Section 3.2) with the following attributes in (2.22):

={0}2J,0={(cc):cJ},+={(cc):cJ},𝒯=(𝕆12S12S𝕆)formulae-sequencesuperscriptsuperscript02𝐽formulae-sequencesuperscript0conditional-setbinomial𝑐𝑐𝑐superscript𝐽formulae-sequencesuperscriptconditional-setbinomial𝑐𝑐𝑐superscript𝐽𝒯𝕆12𝑆12𝑆𝕆\mathcal{R}^{-}=\{0\}^{2J},\ \mathcal{R}^{0}=\left\{\binom{c}{-c}:c\in\mathbb{R}^{J}\right\},\ \mathcal{R}^{+}=\left\{\binom{c}{c}:c\in\mathbb{R}^{J}\right\},\ \mathcal{T}=\left(\begin{array}[c]{cc}\mathbb{O}&\frac{1}{2}S\\ \frac{1}{2}S&\mathbb{O}\end{array}\right) (2.26)

where S=Sh𝑆superscript𝑆S=S^{h} is a symmetric non-degenerate matrix of size J×J𝐽𝐽J\times J. In the next section we will give a different formulation of the abstract equation

𝒜hv=fsuperscript𝒜𝑣𝑓\mathcal{A}^{h}v=f\in\mathcal{L} (2.27)

which will help us to study the spectrum of 𝒜h.superscript𝒜\mathcal{A}^{h}.

2.4 The differential problem with point conditions

Let ±superscriptsubscriptWeierstrass-pplus-or-minus\wp_{\pm}^{\prime} and ±′′superscriptsubscriptWeierstrass-pplus-or-minus′′\wp_{\pm}^{\prime\prime} be projections :J:absentsuperscript𝐽:\mathfrak{H}\rightarrow\mathbb{R}^{J} defined in (2.15). Following [23, 22, 28], we rewrite relations imposed on ±vsuperscriptsubscriptWeierstrass-pplus-or-minus𝑣\wp_{\pm}^{\prime}v and ±′′vsuperscriptsubscriptWeierstrass-pplus-or-minus′′𝑣\wp_{\pm}^{\prime\prime}v in (2.22) according to (2.26), as the point conditions

+′′v+vSvsuperscriptsubscriptWeierstrass-p′′𝑣superscriptsubscriptWeierstrass-p𝑣𝑆superscriptsubscriptWeierstrass-p𝑣\displaystyle\wp_{+}^{\prime\prime}v-\wp_{+}^{\prime}v-S\wp_{-}^{\prime}v =0J,absent0superscript𝐽\displaystyle=0\in\mathbb{R}^{J}, (2.28)
v+′′vsuperscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p′′𝑣\displaystyle\wp_{-}^{\prime}v+\wp_{-}^{\prime\prime}v =0J.absent0superscript𝐽\displaystyle=0\in\mathbb{R}^{J}. (2.29)

We also will deal with the inhomogeneous equation

+′′v+vSv=kJ.superscriptsubscriptWeierstrass-p′′𝑣superscriptsubscriptWeierstrass-p𝑣𝑆superscriptsubscriptWeierstrass-p𝑣𝑘superscript𝐽\wp_{+}^{\prime\prime}v-\wp_{+}^{\prime}v-S\wp_{-}^{\prime}v=k\in\mathbb{R}^{J}. (2.30)

The problems

Δyv0(y)subscriptΔ𝑦subscript𝑣0𝑦\displaystyle-\Delta_{y}v_{0}(y) =f0(y),yω,νv0(y)=0,yωformulae-sequenceabsentsubscript𝑓0𝑦formulae-sequence𝑦subscript𝜔direct-productformulae-sequencesubscript𝜈subscript𝑣0𝑦0𝑦𝜔\displaystyle=f_{0}(y),\ \ y\in\omega_{\odot},\ \ \ \ \partial_{\nu}v_{0}(y)=0,\ \ y\in\partial\omega (2.31)
γj|ωj|z2vj(z)subscript𝛾𝑗subscript𝜔𝑗superscriptsubscript𝑧2subscript𝑣𝑗𝑧\displaystyle-\gamma_{j}|\omega_{j}|\partial_{z}^{2}v_{j}(z) =fj(z),z(0,lj),vj(lj)=0formulae-sequenceabsentsubscript𝑓𝑗𝑧formulae-sequence𝑧0subscript𝑙𝑗subscript𝑣𝑗subscript𝑙𝑗0\displaystyle=f_{j}(z),\ z\in(0,l_{j}),\ \ \ v_{j}(l_{j})=0 (2.32)

with the point conditions (2.29), (2.30) give rise to the continuous mapping

𝔄:={v:v+′′v=0}:=𝔏×J.:𝔄subscriptconditional-set𝑣superscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p′′𝑣0assign𝔏superscript𝐽\mathfrak{A}:\mathfrak{H}_{-}=\{v\in\mathfrak{H}:\wp_{-}^{\prime}v+\wp_{-}^{\prime\prime}v=0\}\rightarrow\mathfrak{R}:=\mathfrak{L}\times\mathbb{R}^{J}. (2.33)
Remark 7

Simple algebraic transformations demonstrate that, under circumstances (2.22) and (2.26), a solution of the equation (2.27) is a solution of the problem (2.31), (2.32), (2.28), (2.29) and vice versa.

Proposition 8

The operator 𝔄𝔄\mathfrak{A} in (2.33) is Fredholm of index zero.

Proof. The point conditions v=0,superscriptsubscriptWeierstrass-p𝑣0\wp_{-}^{\prime}v=0, +′′v=0superscriptsubscriptWeierstrass-p′′𝑣0\wp_{+}^{\prime\prime}v=0 in Remark 6 generate the Fredholm operator of index zero

H2(ω0)×j=1J(H2(Ij)H01(Ij))𝔏.superscript𝐻2subscript𝜔0superscriptsubscriptproduct𝑗1𝐽superscript𝐻2subscript𝐼𝑗superscriptsubscript𝐻01subscript𝐼𝑗𝔏H^{2}(\omega_{0})\times{\textstyle\prod\nolimits_{j=1}^{J}}(H^{2}(I_{j})\cap H_{0}^{1}(I_{j}))\rightarrow\mathfrak{L}. (2.34)

The operator (2.33) is a finite dimensional, i.e. compact, perturbation of (2.34) and thus keeps the Fredholm property. Since S=ST𝑆superscript𝑆𝑇S=S^{T}, the generalized Green formula (2.16) can be written in the symmetric form reflecting the particular point conditions (2.28), (2.29)

q(v,w)𝑞𝑣𝑤\displaystyle q(v,w) =+v+′′v+Sv,wv,+w+′′w+SwabsentsuperscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p′′𝑣𝑆superscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p𝑤superscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p𝑤superscriptsubscriptWeierstrass-p′′𝑤𝑆superscriptsubscriptWeierstrass-p𝑤\displaystyle=\left\langle\wp_{+}^{\prime}v-\wp_{+}^{\prime\prime}v+S\wp_{-}^{\prime}v,\wp_{-}^{\prime}w\right\rangle-\left\langle\wp_{-}^{\prime}v,\wp_{+}^{\prime}w-\wp_{+}^{\prime\prime}w+S\wp_{-}^{\prime}w\right\rangle (2.35)
++′′v,w+′′wv+′′v,+′′wsuperscriptsubscriptWeierstrass-p′′𝑣superscriptsubscriptWeierstrass-p𝑤superscriptsubscriptWeierstrass-p′′𝑤superscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p′′𝑣superscriptsubscriptWeierstrass-p′′𝑤\displaystyle+\left\langle\wp_{+}^{\prime\prime}v,\wp_{-}^{\prime}w+\wp_{-}^{\prime\prime}w\right\rangle-\left\langle\wp_{-}^{\prime}v+\wp_{-}^{\prime\prime}v,\wp_{+}^{\prime\prime}w\right\rangle

and hence an argument in [15, Sect. 2.2.5, 2.5.3], cf. [25, §6.2], shows that

Ind𝔄=dimker𝔄dimcoker𝔄=0,coker𝔄={(v,v):vker𝔄}.\mathrm{Ind~{}}\mathfrak{A}=\dim\ker\mathfrak{A}-\dim\text{{coker}}~{}\mathfrak{A}=0,\ \ \text{{coker}}~{}\mathfrak{A}=\{(v,\wp_{-}^{\prime}v)\in\mathfrak{R}:v\in\ker\mathfrak{A}\}.\ \ \blacksquare (2.36)

Let G𝐺G be the generalized Green function of the Neumann problem (2.24), see, e.g., [32], namely a distributional solution of

ΔyG(y,𝐲)=δ(y𝐲)|ω0|1,yω0,ν(y)G(y,𝐲)=0,yω0,formulae-sequencesubscriptΔ𝑦𝐺𝑦𝐲𝛿𝑦𝐲superscriptsubscript𝜔01formulae-sequence𝑦subscript𝜔0formulae-sequencesubscript𝜈𝑦𝐺𝑦𝐲0𝑦subscript𝜔0\displaystyle-\Delta_{y}G(y,\mathbf{y})=\delta(y-\mathbf{y})-|\omega_{0}|^{-1},\ y\in\omega_{0},\ \ \ \partial_{\nu(y)}G(y,\mathbf{y})=0,\ y\in\partial\omega_{0}, (2.37)
ω0G(y,𝐲)𝑑y=0,𝐲ω0,formulae-sequencesubscriptsubscript𝜔0𝐺𝑦𝐲differential-d𝑦0𝐲subscript𝜔0\displaystyle\int_{\omega_{0}}G(y,\mathbf{y})dy=0,\ \mathbf{y}\in\omega_{0},

where δ𝛿\delta is the Dirac mass. We put Gj(y)=G(y,Pj)superscript𝐺𝑗𝑦𝐺𝑦superscript𝑃𝑗G^{j}(y)=G(y,P^{j}) and write

Gj(y)=χj(y)(2π)1lnrj+G^j(y),G^jH2(ω0).formulae-sequencesuperscript𝐺𝑗𝑦subscript𝜒𝑗𝑦superscript2𝜋1subscript𝑟𝑗superscript^𝐺𝑗𝑦superscript^𝐺𝑗superscript𝐻2subscript𝜔0G^{j}(y)=-\chi_{j}(y)(2\pi)^{-1}\ln r_{j}+\widehat{G}^{j}(y),\ \ \ \widehat{G}^{j}\in H^{2}(\omega_{0}). (2.38)

The J×J𝐽𝐽J\times J-matrix 𝒢𝒢\mathcal{G} with entries 𝒢kj=G^j(Pk)superscriptsubscript𝒢𝑘𝑗superscript^𝐺𝑗superscript𝑃𝑘\mathcal{G}_{k}^{j}=\widehat{G}^{j}(P^{k}) is symmetric, see [4, Sect. 2.2]. We compose the vectors

𝐆j=(Gj,δj1γ11|ω1|1(zl1),,δjJγJ1|ωJ|1(zlJ)),superscript𝐆𝑗superscript𝐺𝑗subscript𝛿𝑗1superscriptsubscript𝛾11superscriptsubscript𝜔11𝑧subscript𝑙1subscript𝛿𝑗𝐽superscriptsubscript𝛾𝐽1superscriptsubscript𝜔𝐽1𝑧subscript𝑙𝐽\mathbf{G}^{j}=(G^{j},\delta_{j1}\gamma_{1}^{-1}|\omega_{1}|^{-1}(z-l_{1}),...,\delta_{jJ}\gamma_{J}^{-1}|\omega_{J}|^{-1}(z-l_{J}))\in\mathfrak{H}, (2.39)

which fall into the subspace ,subscript\mathfrak{H}_{-}, see (2.33), because

𝐆jsuperscriptsubscriptWeierstrass-psuperscript𝐆𝑗\displaystyle\wp_{-}^{\prime}\mathbf{G}^{j} =′′𝐆j=𝐞(j),absentsuperscriptsubscriptWeierstrass-p′′superscript𝐆𝑗subscript𝐞𝑗\displaystyle=-\wp_{-}^{\prime\prime}\mathbf{G}^{j}=\mathbf{e}_{(j)},\ (2.40)
+𝐆jsuperscriptsubscriptWeierstrass-psuperscript𝐆𝑗\displaystyle\wp_{+}^{\prime}\mathbf{G}^{j} =𝒢j=(𝒢1j,,𝒢Jj),+′′𝐆j=γj1|ωj|1lj𝐞(j).formulae-sequenceabsentsuperscript𝒢𝑗superscriptsubscript𝒢1𝑗superscriptsubscript𝒢𝐽𝑗superscriptsubscriptWeierstrass-p′′superscript𝐆𝑗superscriptsubscript𝛾𝑗1superscriptsubscript𝜔𝑗1subscript𝑙𝑗subscript𝐞𝑗\displaystyle=\mathcal{G}^{j}=(\mathcal{G}_{1}^{j},...,\mathcal{G}_{J}^{j}),\ \ \ \wp_{+}^{\prime\prime}\mathbf{G}^{j}=-\gamma_{j}^{-1}|\omega_{j}|^{-1}l_{j}\mathbf{e}_{(j)}.

Here, 𝐞(j)=(δj1,,δjJ),subscript𝐞𝑗subscript𝛿𝑗1subscript𝛿𝑗𝐽\mathbf{e}_{(j)}=(\delta_{j1},...,\delta_{jJ}), j=1,,J,𝑗1𝐽j=1,...,J, is the natural basis in Jsuperscript𝐽\mathbb{R}^{J}.

Let \mathcal{E} be a subspace spanned over the vector ε=(1,,1)J,𝜀11superscript𝐽\varepsilon=(1,...,1)\in\mathbb{R}^{J}, |ε|=J,𝜀𝐽|\varepsilon|=\sqrt{J}, and J=superscript𝐽direct-sumsuperscriptbottom\mathbb{R}^{J}=\mathcal{E\oplus E}^{\bot} with the orthogonal projector 𝒫superscript𝒫bottom\mathcal{P}^{\bot} onto ,superscriptbottom\mathcal{E}^{\bot}, dim=J1dimensionsuperscriptbottom𝐽1\dim\mathcal{E}^{\bot}=J-1. We also introduce the diagonal matrix

𝒬=diag{γ1|ω1|l11,,γJ|ωJ|lJ1}.𝒬diagsubscript𝛾1subscript𝜔1superscriptsubscript𝑙11subscript𝛾𝐽subscript𝜔𝐽superscriptsubscript𝑙𝐽1\mathcal{Q}=\mathrm{diag}\{\gamma_{1}|\omega_{1}|l_{1}^{-1},...,\gamma_{J}|\omega_{J}|l_{J}^{-1}\}. (2.41)
Theorem 9

If the operator

𝒫(S+𝒢+𝒬1)𝒫::superscript𝒫bottom𝑆𝒢superscript𝒬1superscript𝒫bottomsuperscriptbottomsuperscriptbottom\mathcal{P}^{\bot}(S+\mathcal{G}+\mathcal{Q}^{-1})\mathcal{P}^{\bot}:\mathcal{E}^{\bot}\rightarrow\mathcal{E}^{\bot} (2.42)

is invertible, then the problem (2.29)-(2.32) on the hybrid domain Ξ0=ωj=1J(PjIj)superscriptΞ0subscript𝜔direct-productsuperscriptsubscript𝑗1𝐽superscript𝑃𝑗subscript𝐼𝑗\Xi^{0}=\omega_{\odot}\cup{\textstyle\bigcup\nolimits_{j=1}^{J}}(P^{j}\cup I_{j}), fig. 2,a, has a unique solution v𝑣subscriptv\in\mathfrak{H}_{-} for any {f,k}𝑓𝑘\{f,k\}\in\mathfrak{R}. In other words, the operator (2.33) is an isomorphism.

Proof. We search for a solution of the problem in the form

v=𝐯+α1𝐆1++αJ𝐆J𝑣𝐯subscript𝛼1superscript𝐆1subscript𝛼𝐽superscript𝐆𝐽v=\mathbf{v}+\alpha_{1}\mathbf{G}^{1}+...+\alpha_{J}\mathbf{G}^{J} (2.43)

where α=(α1,,αJ)J,𝛼subscript𝛼1subscript𝛼𝐽superscript𝐽\alpha=(\alpha_{1},...,\alpha_{J})\in\mathbb{R}^{J}, 𝐆jsuperscript𝐆𝑗\mathbf{G}^{j} is given in (2.39) and 𝐯=(𝐯0,𝐯1,,𝐯J)𝐯subscript𝐯0subscript𝐯1subscript𝐯𝐽subscript\mathbf{v}=(\mathbf{v}_{0},\mathbf{v}_{1},...,\mathbf{v}_{J})\in\mathfrak{H}_{-} with

𝐯0(y)=α00+𝐯00(y),𝐯00H2(ω0),ω0𝐯00(y)𝑑y=0,𝐯jH2(Ij).formulae-sequencesubscript𝐯0𝑦superscriptsubscript𝛼00superscriptsubscript𝐯00𝑦formulae-sequencesuperscriptsubscript𝐯00superscript𝐻2subscript𝜔0formulae-sequencesubscriptsubscript𝜔0superscriptsubscript𝐯00𝑦differential-d𝑦0subscript𝐯𝑗superscript𝐻2subscript𝐼𝑗\mathbf{v}_{0}(y)=\alpha_{0}^{0}+\mathbf{v}_{0}^{0}(y),\ \mathbf{v}_{0}^{0}\in H^{2}(\omega_{0}),\ \int_{\omega_{0}}\mathbf{v}_{0}^{0}(y)dy=0,\ \mathbf{v}_{j}\in H^{2}(I_{j}). (2.44)

In view of (2.37)-(2.39) these functions must satisfy the problems

Δy𝐯0(y)subscriptΔ𝑦subscript𝐯0𝑦\displaystyle-\Delta_{y}\mathbf{v}_{0}(y) =f0(y)|ω0|1(α1++αJ),yω0,ν𝐯0(y)=0,yω0,formulae-sequenceabsentsubscript𝑓0𝑦superscriptsubscript𝜔01subscript𝛼1subscript𝛼𝐽formulae-sequence𝑦subscript𝜔0formulae-sequencesubscript𝜈subscript𝐯0𝑦0𝑦subscript𝜔0\displaystyle=f_{0}(y)-|\omega_{0}|^{-1}(\alpha_{1}+...+\alpha_{J}),\ \ y\in\omega_{0},\ \ \ \ \partial_{\nu}\mathbf{v}_{0}(y)=0,\ \ y\in\omega_{0}, (2.45)
γj|ωj|z2𝐯j(z)subscript𝛾𝑗subscript𝜔𝑗superscriptsubscript𝑧2subscript𝐯𝑗𝑧\displaystyle-\gamma_{j}|\omega_{j}|\partial_{z}^{2}\mathbf{v}_{j}(z) =fj(z),z(0,lj),𝐯j(lj)=0,γj|ωj|z𝐯j(0)=0.formulae-sequenceabsentsubscript𝑓𝑗𝑧formulae-sequence𝑧0subscript𝑙𝑗formulae-sequencesubscript𝐯𝑗subscript𝑙𝑗0subscript𝛾𝑗subscript𝜔𝑗subscript𝑧subscript𝐯𝑗00\displaystyle=f_{j}(z),\ z\in(0,l_{j}),\ \ \ \ \mathbf{v}_{j}(l_{j})=0,\ \ \ -\gamma_{j}|\omega_{j}|\partial_{z}\mathbf{v}_{j}(0)=0.

Under the condition

jαj=|ω0|ω0f0(y)𝑑y,subscript𝑗subscript𝛼𝑗subscript𝜔0subscriptsubscript𝜔0subscript𝑓0𝑦differential-d𝑦{\textstyle\sum\nolimits_{j}}\alpha_{j}=|\omega_{0}|\int_{\omega_{0}}f_{0}(y)dy, (2.46)

the problems (2.45) have unique solutions (2.44) but with arbitrary constant α00superscriptsubscript𝛼00\alpha_{0}^{0}. The vector function (2.43) fulfils the point condition (2.29) while (2.30) turns into

𝒢αα00ε𝒬1αSα=k++(𝐯00,0,,0)+′′(0,𝐯1,,𝐯J)J.𝒢𝛼superscriptsubscript𝛼00𝜀superscript𝒬1𝛼𝑆𝛼𝑘superscriptsubscriptWeierstrass-psuperscriptsubscript𝐯0000superscriptsubscriptWeierstrass-p′′0subscript𝐯1subscript𝐯𝐽superscript𝐽-\mathcal{G}\alpha-\alpha_{0}^{0}\varepsilon-\mathcal{Q}^{-1}\alpha-S\alpha=k+\wp_{+}^{\prime}(\mathbf{v}_{0}^{0},0,...,0)-\wp_{+}^{\prime\prime}(0,\mathbf{v}_{1},...,\mathbf{v}_{J})\in\mathbb{R}^{J}. (2.47)

Applying the projector 𝒫superscript𝒫bottom\mathcal{P}^{\bot}, we annul the term α00εsuperscriptsubscript𝛼00𝜀\alpha_{0}^{0}\varepsilon in (2.47) and determine 𝒫αsuperscript𝒫bottom𝛼\mathcal{P}^{\bot}\alpha, thanks to our assumption on the mapping (2.42). Then the equation (2.46) gives the remaining part of the coefficient vector in (2.43). Recalling (2.47) yields a value of α00superscriptsubscript𝛼00\alpha_{0}^{0}.

Since we have found a solution (2.43), the operator (2.33) is an epimorphism and becomes isomorphism by virtue of Proposition 8. \blacksquare

2.5 The variational formulation of the problem with point conditions

Similarly to [23, 22, 28] we associate the problem (2.29)-(2.32) with the quadratic form

𝔈(v;f,k)𝔈𝑣𝑓𝑘\displaystyle\mathfrak{E}(v;f,k) =12(Δyv0,v0)ω0(f0,v0)ω0j(γj|ωj|(z2vj,vj)Ij+(fj,vj)Ij\displaystyle=-\frac{1}{2}(\Delta_{y}v_{0},v_{0})_{\omega_{0}}-(f_{0},v_{0})_{\omega_{0}}-\sum\nolimits_{j}(\gamma_{j}|\omega_{j}|(\partial_{z}^{2}v_{j},v_{j})_{I_{j}}+(f_{j},v_{j})_{I_{j}} (2.48)
+12+′′v+vSv,vk,v12superscriptsubscriptWeierstrass-p′′𝑣superscriptsubscriptWeierstrass-p𝑣𝑆superscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p𝑣𝑘superscriptsubscriptWeierstrass-p𝑣\displaystyle+\frac{1}{2}\left\langle\wp_{+}^{\prime\prime}v-\wp_{+}^{\prime}v-S\wp_{-}^{\prime}v,\wp_{-}^{\prime}v\right\rangle-\left\langle k,\wp_{-}^{\prime}v\right\rangle

defined properly in the subspace subscript\mathfrak{H}_{-} of the Hilbert space (2.13). We call (2.48) an energy functional for the problem with point conditions.

Remark 10

In the case k=0𝑘0k=0 the form 𝔈(v;f,0)𝔈𝑣𝑓0\mathfrak{E}(v;f,0) restricted onto 𝒟(𝒜)××𝒟𝒜subscript\mathcal{D}(\mathcal{A})\times\mathcal{L}\subset\mathfrak{H}_{-}\times\mathcal{L} coincides with the energy functional

12(𝒜v,v)(f,v)12subscript𝒜𝑣𝑣subscript𝑓𝑣\frac{1}{2}(\mathcal{A}v,v)_{\mathcal{L}}-(f,v)_{\mathcal{L}}

generated by the self-adjoint extension 𝒜𝒜\mathcal{A} with the parameters (2.26) in Proposition 4. This follows from the fact that two scalar products in Jsuperscript𝐽\mathbb{R}^{J} on the right-hand side of (2.48) vanish.

Theorem 11

A vector function v𝑣v\in\mathfrak{H} is a solution of the problem (2.29)-(2.32) if and only if v𝑣v is a stationary point of the energy functional (2.48).

Proof. Calculating the variation of the functional (2.48), we obtain

δ𝔈(v,w;f,k)𝛿𝔈𝑣𝑤𝑓𝑘\displaystyle\delta\mathfrak{E}(v,w;f,k) =12(Δyv0,w0)ω012(Δyw0,v0)ω0(f0,v0)ω0absent12subscriptsubscriptΔ𝑦subscript𝑣0subscript𝑤0subscript𝜔012subscriptsubscriptΔ𝑦subscript𝑤0subscript𝑣0subscript𝜔0subscriptsubscript𝑓0subscript𝑣0subscript𝜔0\displaystyle=-\frac{1}{2}(\Delta_{y}v_{0},w_{0})_{\omega_{0}}-\frac{1}{2}(\Delta_{y}w_{0},v_{0})_{\omega_{0}}-(f_{0},v_{0})_{\omega_{0}}
j(12γj|ωj|(z2vj,wj)Ij+12γj|ωj|(z2wj,vj)Ij+(fj,wj)Ij)subscript𝑗12subscript𝛾𝑗subscript𝜔𝑗subscriptsuperscriptsubscript𝑧2subscript𝑣𝑗subscript𝑤𝑗subscript𝐼𝑗12subscript𝛾𝑗subscript𝜔𝑗subscriptsuperscriptsubscript𝑧2subscript𝑤𝑗subscript𝑣𝑗subscript𝐼𝑗subscriptsubscript𝑓𝑗subscript𝑤𝑗subscript𝐼𝑗\displaystyle-\sum\nolimits_{j}\left(\frac{1}{2}\gamma_{j}|\omega_{j}|(\partial_{z}^{2}v_{j},w_{j})_{I_{j}}+\frac{1}{2}\gamma_{j}|\omega_{j}|(\partial_{z}^{2}w_{j},v_{j})_{I_{j}}+(f_{j},w_{j})_{I_{j}}\right)
+12+′′v+vSv,w+12+′′w+wSw,vk,w.12superscriptsubscriptWeierstrass-p′′𝑣superscriptsubscriptWeierstrass-p𝑣𝑆superscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p𝑤12superscriptsubscriptWeierstrass-p′′𝑤superscriptsubscriptWeierstrass-p𝑤𝑆superscriptsubscriptWeierstrass-p𝑤superscriptsubscriptWeierstrass-p𝑣𝑘superscriptsubscriptWeierstrass-p𝑤\displaystyle+\frac{1}{2}\left\langle\wp_{+}^{\prime\prime}v-\wp_{+}^{\prime}v-S\wp_{-}^{\prime}v,\wp_{-}^{\prime}w\right\rangle+\frac{1}{2}\left\langle\wp_{+}^{\prime\prime}w-\wp_{+}^{\prime}w-S\wp_{-}^{\prime}w,\wp_{-}^{\prime}v\right\rangle-\left\langle k,\wp_{-}^{\prime}w\right\rangle.

We make use of the generalized Green formula (2.16) while interchanging positions of v𝑣v and w𝑤w. Recalling the relation S=S𝑆superscript𝑆topS=S^{\top} and the point condition (2.29) for v,w𝑣𝑤subscriptv,w\in\mathfrak{H}_{-}, we have

δ𝔈(v,w;f,k)𝛿𝔈𝑣𝑤𝑓𝑘\displaystyle\delta\mathfrak{E}(v,w;f,k) =12(Δyv0,w0)ω012(w0,Δyv0)ω0(f0,v0)ω0absent12subscriptsubscriptΔ𝑦subscript𝑣0subscript𝑤0subscript𝜔012subscriptsubscript𝑤0subscriptΔ𝑦subscript𝑣0subscript𝜔0subscriptsubscript𝑓0subscript𝑣0subscript𝜔0\displaystyle=-\frac{1}{2}(\Delta_{y}v_{0},w_{0})_{\omega_{0}}-\frac{1}{2}(w_{0},\Delta_{y}v_{0})_{\omega_{0}}-(f_{0},v_{0})_{\omega_{0}} (2.49)
j(12γj|ωj|(z2vj,wj)Ij+12γj|ωj|(wj,z2vj)Ij+(fj,wj)Ij)subscript𝑗12subscript𝛾𝑗subscript𝜔𝑗subscriptsuperscriptsubscript𝑧2subscript𝑣𝑗subscript𝑤𝑗subscript𝐼𝑗12subscript𝛾𝑗subscript𝜔𝑗subscriptsubscript𝑤𝑗superscriptsubscript𝑧2subscript𝑣𝑗subscript𝐼𝑗subscriptsubscript𝑓𝑗subscript𝑤𝑗subscript𝐼𝑗\displaystyle-\sum\nolimits_{j}\left(\frac{1}{2}\gamma_{j}|\omega_{j}|(\partial_{z}^{2}v_{j},w_{j})_{I_{j}}+\frac{1}{2}\gamma_{j}|\omega_{j}|(w_{j},\partial_{z}^{2}v_{j})_{I_{j}}+(f_{j},w_{j})_{I_{j}}\right)
+12+w,v+12w,+v+12+′′v+v,w12subscriptWeierstrass-p𝑤subscriptWeierstrass-p𝑣12subscriptWeierstrass-p𝑤subscriptWeierstrass-p𝑣12superscriptsubscriptWeierstrass-p′′𝑣superscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p𝑤\displaystyle+\frac{1}{2}\left\langle\wp_{+}w,\wp_{-}v\right\rangle+\frac{1}{2}\left\langle\wp_{-}w,\wp_{+}v\right\rangle+\frac{1}{2}\left\langle\wp_{+}^{\prime\prime}v-\wp_{+}^{\prime}v,\wp_{-}^{\prime}w\right\rangle
+12+′′w+w,v+12Sv,w12w,Svk,w12superscriptsubscriptWeierstrass-p′′𝑤superscriptsubscriptWeierstrass-p𝑤superscriptsubscriptWeierstrass-p𝑣12𝑆superscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p𝑤12superscriptsubscriptWeierstrass-p𝑤𝑆superscriptsubscriptWeierstrass-p𝑣𝑘superscriptsubscriptWeierstrass-p𝑤\displaystyle+\frac{1}{2}\left\langle\wp_{+}^{\prime\prime}w-\wp_{+}^{\prime}w,\wp_{-}^{\prime}v\right\rangle+\frac{1}{2}\left\langle S\wp_{-}^{\prime}v,\wp_{-}^{\prime}w\right\rangle-\frac{1}{2}\left\langle\wp_{-}^{\prime}w,S\wp_{-}^{\prime}v\right\rangle-\left\langle k,\wp_{-}^{\prime}w\right\rangle
=(Δyv0f0,w0)ω0+j(γj|ωj|z2vjfj,wj)Ij++′′v+vSv,w.absentsubscriptsubscriptΔ𝑦subscript𝑣0subscript𝑓0subscript𝑤0subscript𝜔0subscript𝑗subscriptsubscript𝛾𝑗subscript𝜔𝑗superscriptsubscript𝑧2subscript𝑣𝑗subscript𝑓𝑗subscript𝑤𝑗subscript𝐼𝑗superscriptsubscriptWeierstrass-p′′𝑣superscriptsubscriptWeierstrass-p𝑣𝑆superscriptsubscriptWeierstrass-p𝑣superscriptsubscriptWeierstrass-p𝑤\displaystyle=(-\Delta_{y}v_{0}-f_{0},w_{0})_{\omega_{0}}+\sum\nolimits_{j}(-\gamma_{j}|\omega_{j}|\partial_{z}^{2}v_{j}-f_{j},w_{j})_{I_{j}}+\left\langle\wp_{+}^{\prime\prime}v-\wp_{+}^{\prime}v-S\wp_{-}^{\prime}v,\wp_{-}^{\prime}w\right\rangle.

Here we, in particular, used the relation (2.29). It also should be mentioned that all functions are real as well as the matrix S𝑆S and the vector k𝑘k.

We see that a solution v𝑣v\in\mathfrak{H} of the problem (2.29)-(2.32), annuls the variation (2.49) of the functional (2.48). On the other hand, for any test vector w𝑤subscriptw\in\mathfrak{H}_{-}, the expression with the stationary point v𝑣subscriptv\in\mathfrak{H}_{-} of 𝔈𝔈\mathfrak{E} vanishes; in particular, taking wCc(ω0)×Cc(I1)××Cc(IJ)𝑤superscriptsubscript𝐶𝑐subscript𝜔0superscriptsubscript𝐶𝑐subscript𝐼1superscriptsubscript𝐶𝑐subscript𝐼𝐽w\in C_{c}^{\infty}(\omega_{0})\times C_{c}^{\infty}(I_{1})\times...\times C_{c}^{\infty}(I_{J}) brings the differential equations in (2.31) and (2.32). Thus, the last scalar product in (2.49) is null and (2.30) is fulfilled because =J.superscriptsubscriptWeierstrass-psubscriptsuperscript𝐽\wp_{-}^{\prime}\mathfrak{H}_{-}=\mathbb{R}^{J}. It remains to mention that the boundary conditions (1.22), (1.23) and the point condition (2.29) are kept in the space subscript\mathfrak{H}_{-}. \blacksquare

3 Determination of parameters of an appropriate hybrid model

3.1 The boundary layer phenomenon

An asymptotic analysis performed in [4] gave a detailed description of the behavior of solutions to the stationary problem in Ξ(h)Ξ\Xi(h) near the junction zones. The internal constitution of the boundary layers which appear in the vicinity of the sockets θjh=ωjh×(0,h)superscriptsubscript𝜃𝑗superscriptsubscript𝜔𝑗0\theta_{j}^{h}=\omega_{j}^{h}\times(0,h) and are written in the rapid variables

ξj=(ηj,ζj),ηj=h1(yPj),ζj=h1z,formulae-sequencesuperscript𝜉𝑗superscript𝜂𝑗superscript𝜁𝑗formulae-sequencesuperscript𝜂𝑗superscript1𝑦superscript𝑃𝑗superscript𝜁𝑗superscript1𝑧\xi^{j}=(\eta^{j},\zeta^{j}),\ \ \eta^{j}=h^{-1}(y-P^{j}),\ \ \zeta^{j}=h^{-1}z, (3.1)

depends crucially on the exponent α𝛼\alpha in (1.13). In the case α=1𝛼1\alpha=1, see (1.14), the transmission conditions (1.11), (1.12) decouple and the coordinate dilation leads to two independent limit problems in the semi-infinite cylinder Qj=ωj×+subscript𝑄𝑗subscript𝜔𝑗superscriptQ_{j}=\omega_{j}\times\mathbb{R}^{+} and the perforated layer Λj=(2ω¯j)×(0,1)subscriptΛ𝑗superscript2subscript¯𝜔𝑗01\Lambda_{j}=(\mathbb{R}^{2}\setminus\overline{\omega}_{j})\times(0,1). In [4, Sect. 2.4], we have examined these problems, namely the Neumann problem

γjΔξWj(ξ)subscript𝛾𝑗subscriptΔ𝜉subscript𝑊𝑗𝜉\displaystyle-\gamma_{j}\Delta_{\xi}W_{j}\left(\xi\right) =0,ξQj,γjνWj(ξ)=gj(ξ),ξωj×+,formulae-sequenceabsent0formulae-sequence𝜉subscript𝑄𝑗formulae-sequencesubscript𝛾𝑗subscript𝜈subscript𝑊𝑗𝜉subscript𝑔𝑗𝜉𝜉subscript𝜔𝑗subscript\displaystyle=0,\ \xi\in Q_{j},\ \ \gamma_{j}\partial_{\nu}W_{j}\left(\xi\right)=g_{j}\left(\xi\right),\ \ \ \xi\in\partial\omega_{j}\times\mathbb{R}_{+}, (3.2)
γjζWj(η,0)subscript𝛾𝑗subscript𝜁subscript𝑊𝑗𝜂0\displaystyle-\gamma_{j}\partial_{\zeta}W_{j}\left(\eta,0\right) =0,ηωj,formulae-sequenceabsent0𝜂subscript𝜔𝑗\displaystyle=0,\ \ \eta\in\omega_{j},

where νsubscript𝜈\partial_{\nu} is the outward normal derivative, and the mixed boundary-value problem

ΔξW0(ξ)subscriptΔ𝜉subscript𝑊0𝜉\displaystyle-\Delta_{\xi}W_{0}\left(\xi\right) =0,ξΛj,ζW0(η,0)=ζW0(η,1)=0,η2ω¯j,formulae-sequenceformulae-sequenceabsent0formulae-sequence𝜉subscriptΛ𝑗subscript𝜁subscript𝑊0𝜂0subscript𝜁subscript𝑊0𝜂10𝜂superscript2subscript¯𝜔𝑗\displaystyle=0,\ \xi\in\Lambda_{j},\ \ -\partial_{\zeta}W_{0}\left(\eta,0\right)=\partial_{\zeta}W_{0}\left(\eta,1\right)=0,\ \ \ \eta\in\mathbb{R}^{2}\setminus\overline{\omega}_{j}, (3.3)
W0(ξ)subscript𝑊0𝜉\displaystyle W_{0}\left(\xi\right) =g0(ξ),ξωj×(0,1).formulae-sequenceabsentsubscript𝑔0𝜉𝜉subscript𝜔𝑗01\displaystyle=g_{0}\left(\xi\right),\ \ \xi\in\partial\omega_{j}\times(0,1).

We now point out several special solutions of these problems, that we need in the sequel. First of all, the homogeneous (gj=0subscript𝑔𝑗0g_{j}=0) problem (3.2) has a constant solution, say 𝐰j(ξ)=1subscript𝐰𝑗𝜉1\mathbf{w}_{j}(\xi)=1, and the problem (3.3) with g0(ξ)=1subscript𝑔0𝜉1g_{0}(\xi)=1 is also satisfied by 𝐰j0(ξ)=1superscriptsubscript𝐰𝑗0𝜉1\mathbf{w}_{j}^{0}(\xi)=1. The homogeneous (g0=0subscript𝑔00g_{0}=0) problem (3.3) admits a solution with the logarithmic growth at infinity

𝐖j0(η)=(2π)1(ln|η|+lnclog(ωj))+𝐖~j0(η),𝐖~j0(η)=O(|η|1),|ξ|+,formulae-sequencesuperscriptsubscript𝐖𝑗0𝜂superscript2𝜋1𝜂subscript𝑐subscript𝜔𝑗superscriptsubscript~𝐖𝑗0𝜂formulae-sequencesuperscriptsubscript~𝐖𝑗0𝜂𝑂superscript𝜂1𝜉\mathbf{W}_{j}^{0}\left(\eta\right)=(2\pi)^{-1}\left(\ln|\eta|+\ln c_{\log}\left(\omega_{j}\right)\right)+\widetilde{\mathbf{W}}_{j}^{0}\left(\eta\right),\ \ \ \widetilde{\mathbf{W}}_{j}^{0}\left(\eta\right)=O\left(|\eta|^{-1}\right),\ \ |\xi|\rightarrow+\infty, (3.4)

where clog(ωj)subscript𝑐subscript𝜔𝑗c_{\log}\left(\omega_{j}\right) is the logarithmic capacity of the set ω¯j2subscript¯𝜔𝑗superscript2\overline{\omega}_{j}\subset\mathbb{R}^{2}. Note that the function 𝐖j0superscriptsubscript𝐖𝑗0\mathbf{W}_{j}^{0} in (3.4) is independent of ζ𝜁\zeta and is called the logarithmic capacity potential, see [30, 12]. Finally, according to [4, Lemma 6],

1=ωjgj(η)𝑑sη,gj(η)=ν𝐖j0(η)formulae-sequence1subscriptsubscript𝜔𝑗subscript𝑔𝑗𝜂differential-dsubscript𝑠𝜂subscript𝑔𝑗𝜂subscript𝜈superscriptsubscript𝐖𝑗0𝜂-1={\displaystyle\int_{\partial\omega_{j}}}g_{j}\left(\eta\right)ds_{\eta},\ \ \ g_{j}\left(\eta\right)=\partial_{\nu}\mathbf{W}_{j}^{0}\left(\eta\right) (3.5)

and a solution of the Neumann problem (3.2) with the datum in (3.5) can be found in the form

𝐖j(η,ζ)=γj1|ωj|1ζ+O(eδζ),δ>0.formulae-sequencesubscript𝐖𝑗𝜂𝜁superscriptsubscript𝛾𝑗1superscriptsubscript𝜔𝑗1𝜁𝑂superscript𝑒𝛿𝜁𝛿0\mathbf{W}_{j}\left(\eta,\zeta\right)=\gamma_{j}^{-1}|\omega_{j}|^{-1}\zeta+O(e^{-\delta\zeta}),\ \ \delta>0. (3.6)

3.2 Individual choice of the self-adjoint extension

Applying the method of matched asymptotic expansions, see for example [33, 11], [16, Ch. 2], we take some functions v00,subscript𝑣0subscript0v_{0}\in\mathfrak{H}_{0}, vjjsubscript𝑣𝑗subscript𝑗v_{j}\in\mathfrak{H}_{j} and write the outer expansions in the plate Ω0(h)subscriptΩ0\Omega_{0}(h) and the rod Ωj(h)subscriptΩ𝑗\Omega_{j}(h) near but outside the socket θjhsuperscriptsubscript𝜃𝑗\theta_{j}^{h}

v0(y)subscript𝑣0𝑦\displaystyle v_{0}(y) =bj2πln1rj+v^0(Pj)+=bj2πln1|ηj|bj2πlnh+v^0(Pj)+,absentsubscript𝑏𝑗2𝜋1subscript𝑟𝑗subscript^𝑣0superscript𝑃𝑗subscript𝑏𝑗2𝜋1subscript𝜂𝑗subscript𝑏𝑗2𝜋subscript^𝑣0superscript𝑃𝑗\displaystyle=\frac{b_{j}}{2\pi}\ln\frac{1}{r_{j}}+\widehat{v}_{0}(P^{j})+...=\frac{b_{j}}{2\pi}\ln\frac{1}{|\eta_{j}|}-\frac{b_{j}}{2\pi}\ln h+\widehat{v}_{0}(P^{j})+..., (3.7)
vj(z)subscript𝑣𝑗𝑧\displaystyle v_{j}(z) =vj(0)+zzvj(0)+=vj(0)+hζjvj(0)+absentsubscript𝑣𝑗0𝑧subscript𝑧subscript𝑣𝑗0subscript𝑣𝑗0superscript𝜁𝑗subscript𝑣𝑗0\displaystyle=v_{j}(0)+z\partial_{z}v_{j}(0)+...=v_{j}(0)+h\zeta^{j}v_{j}(0)+... (3.8)

Here, ellipses stand for higher order terms of no importance in our asymptotic procedure. The inner expansions in the immediate vicinity of the sockets are composed from the above described solutions of the problems (3.3) and (3.2)

cj0𝐰j0(ξj)+cj1𝐖j0(ξj)superscriptsubscript𝑐𝑗0superscriptsubscript𝐰𝑗0superscript𝜉𝑗superscriptsubscript𝑐𝑗1superscriptsubscript𝐖𝑗0superscript𝜉𝑗\displaystyle c_{j}^{0}\mathbf{w}_{j}^{0}(\xi^{j})+c_{j}^{1}\mathbf{W}_{j}^{0}(\xi^{j}) =cj112πln1|ηj|+cj112πlnclog(ωj)+cj0+,(Pj+hηj,hζj)Ω(h),formulae-sequenceabsentsuperscriptsubscript𝑐𝑗112𝜋1subscript𝜂𝑗superscriptsubscript𝑐𝑗112𝜋subscript𝑐subscript𝜔𝑗superscriptsubscript𝑐𝑗0superscript𝑃𝑗superscript𝜂𝑗superscript𝜁𝑗subscriptΩ\displaystyle=c_{j}^{1}\frac{1}{2\pi}\ln\frac{1}{|\eta_{j}|}+c_{j}^{1}\frac{1}{2\pi}\ln c_{\log}\left(\omega_{j}\right)+c_{j}^{0}+...,\ (P^{j}+h\eta^{j},h\zeta^{j})\in\Omega_{\bullet}(h), (3.9)
cj0𝐰j(ξj)+hcj1𝐖j(ξj)superscriptsubscript𝑐𝑗0subscript𝐰𝑗superscript𝜉𝑗superscriptsubscript𝑐𝑗1subscript𝐖𝑗superscript𝜉𝑗\displaystyle c_{j}^{0}\mathbf{w}_{j}(\xi^{j})+hc_{j}^{1}\mathbf{W}_{j}(\xi^{j}) =hcj1γj1|ωj|1ζj+cj0+,(Pj+hηj,hζj)Ωj(h).formulae-sequenceabsentsuperscriptsubscript𝑐𝑗1superscriptsubscript𝛾𝑗1superscriptsubscript𝜔𝑗1superscript𝜁𝑗superscriptsubscript𝑐𝑗0superscript𝑃𝑗superscript𝜂𝑗superscript𝜁𝑗subscriptΩ𝑗\displaystyle=hc_{j}^{1}\gamma_{j}^{-1}|\omega_{j}|^{-1}\zeta^{j}+c_{j}^{0}+...,\ \ \ (P^{j}+h\eta^{j},h\zeta^{j})\in\Omega_{j}(h). (3.10)

We emphasize that, by definition of those solutions, the transmission condition (1.12) is wholly satisfied by (3.9) and (3.10) but the condition (1.12) with the reasonable precision O(h).𝑂O(h).

Comparing (3.7) with (3.9) and (3.8) with (3.10) yields the equations

bjsubscript𝑏𝑗\displaystyle b_{j} =cj1,v^0(Pj)bj(2π)1lnh=cj0+cj1(2π)1lnclog(ωj),formulae-sequenceabsentsuperscriptsubscript𝑐𝑗1subscript^𝑣0superscript𝑃𝑗subscript𝑏𝑗superscript2𝜋1superscriptsubscript𝑐𝑗0superscriptsubscript𝑐𝑗1superscript2𝜋1subscript𝑐subscript𝜔𝑗\displaystyle=c_{j}^{1},\ \ \widehat{v}_{0}(P^{j})-b_{j}(2\pi)^{-1}\ln h=c_{j}^{0}+c_{j}^{1}(2\pi)^{-1}\ln c_{\log}\left(\omega_{j}\right), (3.11)
zvj(0)subscript𝑧subscript𝑣𝑗0\displaystyle\partial_{z}v_{j}(0) =cj1γj1|ωj|1,vj(0)=cj0.formulae-sequenceabsentsuperscriptsubscript𝑐𝑗1superscriptsubscript𝛾𝑗1superscriptsubscript𝜔𝑗1subscript𝑣𝑗0superscriptsubscript𝑐𝑗0\displaystyle=c_{j}^{1}\gamma_{j}^{-1}|\omega_{j}|^{-1},\ \ v_{j}(0)=c_{j}^{0}.

Excluding the coefficients cj1superscriptsubscript𝑐𝑗1c_{j}^{1} and cj0superscriptsubscript𝑐𝑗0c_{j}^{0}, we derive the relations

bjγj|ωj|zvj(0)subscript𝑏𝑗subscript𝛾𝑗subscript𝜔𝑗subscript𝑧subscript𝑣𝑗0\displaystyle b_{j}-\gamma_{j}|\omega_{j}|\partial_{z}v_{j}(0) =0,absent0\displaystyle=0, (3.12)
v^0(Pj)vj(0)subscript^𝑣0superscript𝑃𝑗subscript𝑣𝑗0\displaystyle\widehat{v}_{0}(P^{j})-v_{j}(0) =bj(2π)1(lnh+lnclog(ωj)).absentsubscript𝑏𝑗superscript2𝜋1subscript𝑐subscript𝜔𝑗\displaystyle=b_{j}(2\pi)^{-1}(\ln h+\ln c_{\log}\left(\omega_{j}\right)).

We also introduce the diagonal J×J𝐽𝐽J\times J-matrix

Sh=(2π)1diag{lnh+lnclog(ω1),,lnh+lnclog(ωJ)}=(2π)1(|lnh|𝕀+Clog)superscript𝑆superscript2𝜋1diagsubscript𝑐subscript𝜔1subscript𝑐subscript𝜔𝐽superscript2𝜋1𝕀subscript𝐶S^{h}=-(2\pi)^{-1}\mathrm{diag}\{\ln h+\ln c_{\log}\left(\omega_{1}\right),...,\ln h+\ln c_{\log}\left(\omega_{J}\right)\}=(2\pi)^{-1}(|\ln h|\mathbb{I}+C_{\log}) (3.13)

which is positive definite for h(0,h0)0subscript0h\in(0,h_{0}) with a small h0(0,1)subscript001h_{0}\in(0,1) and further substitutes for S𝑆S in (2.26). Here, 𝕀𝕀\mathbb{I} is the unit J×J𝐽𝐽J\times J-matrix and Clog=diag{lnclog(ω1),,lnclog(ωJ)}subscript𝐶diagsubscript𝑐subscript𝜔1subscript𝑐subscript𝜔𝐽C_{\log}=\mathrm{diag}\{\ln c_{\log}\left(\omega_{1}\right),...,\ln c_{\log}\left(\omega_{J}\right)\}.

The main conclusion from the above consideration is that the relations (3.12) between the functions v00subscript𝑣0subscript0v_{0}\in\mathfrak{H}_{0} and vjjsubscript𝑣𝑗subscript𝑗v_{j}\in\mathfrak{H}_{j}, J=1,,J𝐽1𝐽J=1,...,J are equivalent to conditions in (2.22), (2.26) defining a self-adjoint extension 𝒜hsuperscript𝒜\mathcal{A}^{h} of the operator A=(A0,A1,,AJ)𝐴subscript𝐴0subscript𝐴1subscript𝐴𝐽A=(A_{0},A_{1},...,A_{J}) in the space 𝔏𝔏\mathfrak{L}, see (2.14), examined in Section 2. In what follows we deal with this operator 𝒜hsuperscript𝒜\mathcal{A}^{h}.

3.3 The spectral problem

Based on results in Section 2, we give two models of the spectral problem (1.6)-(1.12). First, we use the introduced self-adjoint operator 𝒜hsuperscript𝒜\mathcal{A}^{h} and write the equation

𝒜hvh=μhvh in 𝔏.superscript𝒜superscript𝑣superscript𝜇superscript𝑣 in 𝔏\mathcal{A}^{h}v^{h}=\mu^{h}v^{h}\text{ \ in }\mathfrak{L}. (3.14)

Second, we supply the problems (1.20), (1.22) and (1.21), (1.23) with the point conditions (2.28), (2.29) and formulate the abstract equation

𝔄hvh=μh(vh,0) in 𝔏×Jsuperscript𝔄superscript𝑣superscript𝜇superscript𝑣0 in 𝔏superscript𝐽\mathfrak{A}^{h}v^{h}=\mu^{h}(v^{h},0)\text{ \ in }\mathfrak{L\times}\mathbb{R}^{J} (3.15)

where the operator 𝔄h:𝔏×J:superscript𝔄subscript𝔏superscript𝐽\mathfrak{A}^{h}:\mathfrak{H}_{-}\rightarrow\mathfrak{L\times}\mathbb{R}^{J} involves the point condition (2.28) with the operator S=Sh𝑆superscript𝑆S=S^{h} in (3.13) while zero at the last position in (3.15) indicates that this condition is homogeneous, namely k=0𝑘0k=0 in (2.30).

According to Remark 7, the spectral equations (3.14) and (3.15) are equivalent with each other. Unfortunately, the operator 𝒜hsuperscript𝒜\mathcal{A}^{h} intended to model the problem (1.6)-(1.12) with the positive spectrum (1.18), is not positive definite in view of Remark 10 and formula (2.48) involving the negative definite matrix S=Sh𝑆superscript𝑆-S=-S^{h}. However, in Remark 5 we mentioned the positive operator 𝒜0superscript𝒜0\mathcal{A}^{0} with the attributes (2.23) in (2.22) which differs from the operator 𝒜hsuperscript𝒜\mathcal{A}^{h} only in subspace of dimension J𝐽J. Thus, the max-min principle, cf. [2, Thm. 10.2.2] demonstrates that the total multiplicity of the negative part of the spectrum of 𝒜hsuperscript𝒜\mathcal{A}^{h} cannot exceed J𝐽J. In the next section we will construct asymptotics of negative eigenvalues which we call parasitic.

3.4 Asymptotics of parasitic eigenvalues

We will need the fundamental solution ΦΦ\Phi of the operator Δy+1subscriptΔ𝑦1-\Delta_{y}+1 in the plane 2superscript2\mathbb{R}^{2}. Its expression is well known, in particular

Φ(y)=O(eψ|y|) as |y|+,Φ(y)=12πln1|y|+Ψ+O(|y|) as |y|+0formulae-sequenceΦ𝑦𝑂superscript𝑒𝜓𝑦 as 𝑦Φ𝑦12𝜋1𝑦Ψ𝑂𝑦 as 𝑦0\Phi(y)=O(e^{-\psi|y|})\text{ as }|y|\rightarrow+\infty,\ \ \ \Phi(y)=\frac{1}{2\pi}\ln\frac{1}{|y|}+\Psi+O(|y|)\text{ as }|y|\rightarrow+0 (3.16)

but exact values of ψ>0𝜓0\psi>0 and ΨΨ\Psi are of no importance here. Let

μh=h2(e2m+O(h)).superscript𝜇superscript2superscript𝑒2𝑚𝑂\mu^{h}=-h^{-2}(e^{2m}+O(h)). (3.17)

We set

v0h(y)superscriptsubscript𝑣0𝑦\displaystyle v_{0}^{h}(y) =jαjχj(y)Φ(h1em(yPj)),absentsubscript𝑗subscript𝛼𝑗subscript𝜒𝑗𝑦Φsuperscript1superscript𝑒𝑚𝑦superscript𝑃𝑗\displaystyle={\textstyle\sum\nolimits_{j}}\alpha_{j}\chi_{j}(y)\Phi(h^{-1}e^{m}(y-P^{j})), (3.18)
vjh(y)superscriptsubscript𝑣𝑗𝑦\displaystyle v_{j}^{h}(y) =αjXj(z)γj1|ωj|1hemeh1emz.absentsubscript𝛼𝑗subscript𝑋𝑗𝑧superscriptsubscript𝛾𝑗1superscriptsubscript𝜔𝑗1superscript𝑒𝑚superscript𝑒superscript1superscript𝑒𝑚𝑧\displaystyle=\alpha_{j}X_{j}(z)\gamma_{j}^{-1}|\omega_{j}|^{-1}he^{-m}e^{-h^{-1}e^{m}z}. (3.19)

where the column α=(α1,,αJ)J𝛼subscript𝛼1subscript𝛼𝐽superscript𝐽\alpha=(\alpha_{1},...,\alpha_{J})\in\mathbb{R}^{J} and the number m𝑚m\in\mathbb{R} are to be determined and XjC(Ij),subscript𝑋𝑗superscript𝐶subscript𝐼𝑗X_{j}\in C^{\infty}(I_{j}), Xj(z)=1subscript𝑋𝑗𝑧1X_{j}(z)=1 for [0,lj/3]0subscript𝑙𝑗3[0,l_{j}/3] and Xj(z)=0subscript𝑋𝑗𝑧0X_{j}(z)=0 for [2lj/3,lj]2subscript𝑙𝑗3subscript𝑙𝑗[2l_{j}/3,l_{j}]. Clearly, the functions (3.18), (3.19) satisfy the boundary conditions (1.22), (1.23) and leave small discrepancies O(eδ/h),𝑂superscript𝑒𝛿O(e^{-\delta/h}), δ>0,𝛿0\delta>0, in the differential equations (1.20), (1.21) with the spectral parameter h2e2msuperscript2superscript𝑒2𝑚-h^{-2}e^{2m}. It should be mentioned that the exponential decay of the boundary layer terms in (3.18) and (3.19) is due to the negative value of μhsuperscript𝜇\mu^{h} in (3.17).

The vector vh=(v0h,v1h,,vJh)superscript𝑣superscriptsubscript𝑣0superscriptsubscript𝑣1superscriptsubscript𝑣𝐽v^{h}=(v_{0}^{h},v_{1}^{h},...,v_{J}^{h}) has the following projections:

+vhsuperscriptsubscriptWeierstrass-psuperscript𝑣\displaystyle\wp_{+}^{\prime}v^{h} =(2π)1(mlnh)+ϕ0,vh=α,formulae-sequenceabsentsuperscript2𝜋1𝑚subscriptitalic-ϕ0superscriptsubscriptWeierstrass-psuperscript𝑣𝛼\displaystyle=(2\pi)^{-1}(m-\ln h)+\phi_{0},\ \ \wp_{-}^{\prime}v^{h}=\alpha, (3.20)
+′′vhsuperscriptsubscriptWeierstrass-p′′superscript𝑣\displaystyle\wp_{+}^{\prime\prime}v^{h} =Qhem,′′vh=α,formulae-sequenceabsent𝑄superscript𝑒𝑚superscriptsubscriptWeierstrass-p′′superscript𝑣𝛼\displaystyle=Qhe^{-m},\ \ \wp_{-}^{\prime\prime}v^{h}=-\alpha,

where Q=diag{γ11|ω1|1,,γJ1|ωJ|1}.𝑄diagsuperscriptsubscript𝛾11superscriptsubscript𝜔11superscriptsubscript𝛾𝐽1superscriptsubscript𝜔𝐽1Q=\mathrm{diag}\{\gamma_{1}^{-1}|\omega_{1}|^{-1},...,\gamma_{J}^{-1}|\omega_{J}|^{-1}\}. Hence, the point condition (2.29) is fulfilled while, in view of (3.13) and (3.20), the condition (2.28) converts into

Qhemα(2π)1(mlnh)α+(2π)1(lnh𝕀+Clog)α=0𝑄superscript𝑒𝑚𝛼superscript2𝜋1𝑚𝛼superscript2𝜋1𝕀subscript𝐶𝛼0Qhe^{m}\alpha-(2\pi)^{-1}(m-\ln h)\alpha+(2\pi)^{-1}(\ln h\mathbb{I}+C_{\log})\alpha=0

or, what is the same,

Clogα=mα2πhemQα.subscript𝐶𝛼𝑚𝛼2𝜋superscript𝑒𝑚𝑄𝛼C_{\log}\alpha=m\alpha-2\pi he^{-m}Q\alpha. (3.21)

Since both matrices are diagonal, the system (3.21) splits into J𝐽J independent transcendental equations. The small factor hh of the exponent emsuperscript𝑒𝑚e^{-m} allows us to apply the implicit function theorem and to obtain the solutions

mjh=logclog(ωj)+O(h),α(j)h=𝐞(j)+O(h),j=1,,J.formulae-sequencesuperscriptsubscript𝑚𝑗subscript𝑐subscript𝜔𝑗𝑂formulae-sequencesuperscriptsubscript𝛼𝑗subscript𝐞𝑗𝑂𝑗1𝐽m_{j}^{h}=\log c_{\log}(\omega_{j})+O(h),\ \ \ \alpha_{(j)}^{h}=\mathbf{e}_{(j)}+O(h),\ \ \ j=1,...,J. (3.22)

We have derived ”good approximations” for J𝐽J negative eigenvalues of the spectral equations (3.14) and (3.15). Recalling that their number cannot exceed J,𝐽J, we come in position to formulate an assertion on the whole negative part of the spectrum.

Proposition 12

There exist positive numbers hsubscripth_{-} and csubscript𝑐c_{-} such that, for h(0,h]0subscripth\in(0,h_{-}], the equation (3.14) or (3.15) possesses exactly J𝐽J eigenvalues on the semi-axis =(,0).subscript0\mathbb{R}_{-}=(-\infty,0). These eigenvalues obey the asymptotic form

|μjh+h2(clog(ωj))2|ch1superscriptsubscript𝜇𝑗superscript2superscriptsubscript𝑐subscript𝜔𝑗2subscript𝑐superscript1|\mu_{-j}^{h}+h^{-2}(c_{\log}(\omega_{j}))^{2}|\leq c_{-}h^{-1} (3.23)

where clog(ωj)>0subscript𝑐subscript𝜔𝑗0c_{\log}(\omega_{j})>0 is the logarithmic capacity of the set ω¯j2subscript¯𝜔𝑗superscript2\overline{\omega}_{j}\subset\mathbb{R}^{2}, see [30, 12].

We will outline the proof in Remark 17. Notice that the negative eigenvalues μjh=O(h2),superscriptsubscript𝜇𝑗𝑂superscript2\mu_{-j}^{h}=O(h^{-2}), j=1,,J,𝑗1𝐽j=1,...,J, are situated very far away from the positive part of the spectrum, that is, outside the scope of the asymptotic models under consideration.

4 Justification of the asymptotic models

4.1 The first convergence theorem

Let μhsuperscript𝜇\mu^{h} be an eigenvalue of the self-adjoint operator 𝒜hsuperscript𝒜\mathcal{A}^{h} in (3.14). The corresponding eigenvector vh=(v0h,v1h,,vJh)𝒟(𝒜h)superscript𝑣superscriptsubscript𝑣0superscriptsubscript𝑣1superscriptsubscript𝑣𝐽𝒟superscript𝒜subscriptv^{h}=(v_{0}^{h},v_{1}^{h},...,v_{J}^{h})\in\mathcal{D}(\mathcal{A}^{h})\subset\mathfrak{H}_{-} can be normed as follows:

||v0h;L2(ω0)||2+jρj|ωj|||vjh;L2(Ij)||2||v_{0}^{h};L^{2}(\omega_{0})||^{2}+{\textstyle\sum\nolimits_{j}}\rho_{j}|\omega_{j}|~{}||v_{j}^{h};L^{2}(I_{j})||^{2} (4.1)

Assuming that

|μh|c,superscript𝜇𝑐|\mu^{h}|\leq c, (4.2)

in particular, rejecting negative eigenvalues in Proposition 12, we recall the Kondratiev theory used in Section 2.1. Then, a solution v0hHloc2(ω¯0𝒫)L2(ω0)superscriptsubscript𝑣0superscriptsubscript𝐻𝑙𝑜𝑐2subscript¯𝜔0𝒫superscript𝐿2subscript𝜔0v_{0}^{h}\in H_{loc}^{2}(\overline{\omega}_{0}\setminus\mathcal{P})\cap L^{2}(\omega_{0}) of the problem (1.20), (1.22) admits the decomposition (2.9) with the ingredients ajh=v^h(Pj),superscriptsubscript𝑎𝑗superscript^𝑣superscript𝑃𝑗a_{j}^{h}=\widehat{v}^{h}(P^{j}), bjhsuperscriptsubscript𝑏𝑗b_{j}^{h}\in\mathbb{R} and v^0hH2(ω0)H01(ω0)superscriptsubscript^𝑣0superscript𝐻2subscript𝜔0superscriptsubscript𝐻01subscript𝜔0\widehat{v}_{0}^{h}\in H^{2}(\omega_{0})\cap H_{0}^{1}(\omega_{0}) while

||v^0h;H2(ω0)||2+j|bjh|c(1+|μh|)||v0h;L2(ω0)||2C,||\widehat{v}_{0}^{h};H^{2}(\omega_{0})||^{2}+{\textstyle\sum\nolimits_{j}}|b_{j}^{h}|\leq c(1+|\mu^{h}|)||v_{0}^{h};L^{2}(\omega_{0})||^{2}\leq C, (4.3)

where v^0hsuperscriptsubscript^𝑣0\widehat{v}_{0}^{h} is given in (2.11).

Furthermore, a solution vjhL2(Ij)superscriptsubscript𝑣𝑗superscript𝐿2subscript𝐼𝑗v_{j}^{h}\in L^{2}(I_{j}) of the ordinary differential equation (1.21) with the Dirichlet condition (1.23) falls into the Sobolev space H2(Ij)superscript𝐻2subscript𝐼𝑗H^{2}(I_{j}) and fulfills the estimate

|vjh(0)|+|zvjh(0)|c||vjh;H2(Ij)||c(1+|μh|)||v0h;L2(Ij)||2C.|v_{j}^{h}(0)|+|\partial_{z}v_{j}^{h}(0)|\leq c||v_{j}^{h};H^{2}(I_{j})||\leq c(1+|\mu^{h}|)||v_{0}^{h};L^{2}(I_{j})||^{2}\leq C. (4.4)

The inequalities (4.2)-(4.4) help to conclude the following convergence along an infinitesimal positive sequence {hk}ksubscriptsubscript𝑘𝑘\{h_{k}\}_{k\in\mathbb{N}}:

μhsuperscript𝜇\displaystyle\mu^{h} μ0,bh=(b1h,,bJh)b0J,formulae-sequenceabsentsuperscript𝜇0superscript𝑏superscriptsubscript𝑏1superscriptsubscript𝑏𝐽superscript𝑏0superscript𝐽\displaystyle\rightarrow\mu^{0}\in\mathbb{R},\ \ \ b^{h}=(b_{1}^{h},...,b_{J}^{h})\rightarrow b^{0}\in\mathbb{R}^{J},\ (4.5)
v^0hsuperscriptsubscript^𝑣0\displaystyle\widehat{v}_{0}^{h} v^00 weakly in H2(ω0),vjhvj0 weakly in H2(Ij).formulae-sequenceabsentsuperscriptsubscript^𝑣00 weakly in superscript𝐻2subscript𝜔0superscriptsubscript𝑣𝑗superscriptsubscript𝑣𝑗0 weakly in superscript𝐻2subscript𝐼𝑗\displaystyle\rightharpoonup\widehat{v}_{0}^{0}\text{ weakly in }H^{2}(\omega_{0}),\ v_{j}^{h}\rightharpoonup v_{j}^{0}\text{ weakly in }H^{2}(I_{j}).

This and the embeddings H2(ω0)C(ω0),superscript𝐻2subscript𝜔0𝐶subscript𝜔0H^{2}(\omega_{0})\subset C(\omega_{0}), H2(Ij)C1(Ij)superscript𝐻2subscript𝐼𝑗superscript𝐶1subscript𝐼𝑗H^{2}(I_{j})\subset C^{1}(I_{j}) imply the convergence of the projections (2.15)

±vh±v0J.subscriptWeierstrass-pplus-or-minussuperscript𝑣subscriptWeierstrass-pplus-or-minussuperscript𝑣0superscript𝐽\wp_{\pm}v^{h}\rightarrow\wp_{\pm}v^{0}\in\mathbb{R}^{J}. (4.6)

We emphasize that formulas in (4.5) guarantee the strong convergences v0hv00superscriptsubscript𝑣0superscriptsubscript𝑣00v_{0}^{h}\rightarrow v_{0}^{0} in L2(ω0)superscript𝐿2subscript𝜔0L^{2}(\omega_{0}) and vjhvj0superscriptsubscript𝑣𝑗superscriptsubscript𝑣𝑗0v_{j}^{h}\rightarrow v_{j}^{0} in L2(Ij)superscript𝐿2subscript𝐼𝑗L^{2}(I_{j}) so that the normalization condition (4.1) is kept by the limit v0=(v00,v10,,vJ0)superscript𝑣0superscriptsubscript𝑣00superscriptsubscript𝑣10superscriptsubscript𝑣𝐽0v^{0}=(v_{0}^{0},v_{1}^{0},...,v_{J}^{0}). Moreover, the differential equations (1.20), (1.21) with μ=μh𝜇superscript𝜇\mu=\mu^{h} and the boundary conditions (1.22), (1.23) for v0h,superscriptsubscript𝑣0v_{0}^{h}, vjhsuperscriptsubscript𝑣𝑗v_{j}^{h} are passed to the limits

μ0,v00=v^00(2π)1jχjbjlnrj, vjh,j=1,,J.formulae-sequencesuperscript𝜇0superscriptsubscript𝑣00superscriptsubscript^𝑣00superscript2𝜋1subscript𝑗subscript𝜒𝑗subscript𝑏𝑗subscript𝑟𝑗 superscriptsubscript𝑣𝑗𝑗1𝐽\mu^{0},\ \ \ v_{0}^{0}=\widehat{v}_{0}^{0}-(2\pi)^{-1}{\textstyle\sum\nolimits_{j}}\chi_{j}b_{j}\ln r_{j},\text{ \ \ }v_{j}^{h},\ j=1,...,J. (4.7)

In order to formulate the next assertion it suffices to mention that the point conditions (2.28), (2.29) with the matrix (3.13) containing the big component (2π)1lnh𝕀superscript2𝜋1𝕀-(2\pi)^{-1}\ln h\mathbb{I} turn in the limit into the relations

v0=0,v0+′′v0=0v0=′′v0=0J.formulae-sequencesuperscriptsubscriptWeierstrass-psuperscript𝑣00formulae-sequencesuperscriptsubscriptWeierstrass-psuperscript𝑣0superscriptsubscriptWeierstrass-p′′superscript𝑣00superscriptsubscriptWeierstrass-psuperscript𝑣0superscriptsubscriptWeierstrass-p′′superscript𝑣00superscript𝐽\wp_{-}^{\prime}v^{0}=0,\ \ \wp_{-}^{\prime}v^{0}+\wp_{-}^{\prime\prime}v^{0}=0\ \ \Rightarrow\ \ \wp_{-}^{\prime}v^{0}=\wp_{-}^{\prime\prime}v^{0}=0\in\mathbb{R}^{J}. (4.8)

These provide the self-adjoint extension 𝒜0superscript𝒜0\mathcal{A}^{0} with the attributes (2.23) in (2.22) that corresponds to the Neumann and mixed boundary-value problems (2.24) and (2.25). The spectra {ϰn0}nsubscriptsuperscriptsubscriptitalic-ϰ𝑛0𝑛\{\varkappa_{n}^{0}\}_{n\in\mathbb{N}} and {ϰnj=π2lj2γjρj(n+12)2}n,subscriptsuperscriptsubscriptitalic-ϰ𝑛𝑗superscript𝜋2superscriptsubscript𝑙𝑗2subscript𝛾𝑗subscript𝜌𝑗superscript𝑛122𝑛\{\varkappa_{n}^{j}=\frac{\pi^{2}}{l_{j}^{2}}\frac{\gamma_{j}}{\rho_{j}}(n+\frac{1}{2})^{2}\}_{n\in\mathbb{N}}, j=1,,J,𝑗1𝐽j=1,...,J, of the above mentioned problems are united into the common monotone sequence

{μn0}n,μ10=0<μ2.0subscriptsuperscriptsubscript𝜇𝑛0𝑛superscriptsubscript𝜇100superscriptsubscript𝜇20\{\mu_{n}^{0}\}_{n\in\mathbb{N}},\ \ \mu_{1}^{0}=0<\mu_{2.}^{0} (4.9)

Here, eigenvalues are listed while counting their multiplicity in.

Theorem 13

If an eigenvalue μhsuperscript𝜇\mu^{h} of the operator 𝒜hsuperscript𝒜\mathcal{A}^{h}, cf. (3.14) and (3.15), and the corresponding eigenvector vhsuperscript𝑣v^{h} fulfil the requirements (4.2) and (4.1), then the limits μ0superscript𝜇0\mu^{0} and v0superscript𝑣0v^{0} in (4.5) along an infinitesimal sequence {hn}nsubscriptsubscript𝑛𝑛\{h_{n}\}_{n\in\mathbb{N}} are an eigenvalue of the operator 𝒜0superscript𝒜0\mathcal{A}^{0} described in Remark 5 and the corresponding eigenvector normed in the space \mathcal{L}, see (2.14).

4.2 The second convergence theorem

In the next section we will verify that entries of the eigenvalue sequence (1.18) of the original problem (1.6)-(1.12) in the junction Ξ(h)3Ξsuperscript3\Xi(h)\subset\mathbb{R}^{3} satisfy the inequalities

0<λn(h)cn for h(0,hn]0superscript𝜆𝑛subscript𝑐𝑛 for 0subscript𝑛0<\lambda^{n}(h)\leq c_{n}\text{ for }h\in(0,h_{n}] (4.10)

with some positive hnsubscript𝑛h_{n} and cnsubscript𝑐𝑛c_{n} which depend on the eigenvalue number n𝑛n but are independent of hh. The corresponding eigenfunction un(h,)H01(Ξ(h);Γ(h))superscript𝑢𝑛superscriptsubscript𝐻01ΞΓu^{n}(h,\cdot)\in H_{0}^{1}(\Xi(h);\Gamma(h)) is subject to the normalization condition (1.19). We introduce the functions

v0n(h,y)superscriptsubscript𝑣0𝑛𝑦\displaystyle v_{0}^{n}(h,y) =1h0hun(h,y,z)𝑑z,yω0,formulae-sequenceabsent1superscriptsubscript0superscript𝑢𝑛𝑦𝑧differential-d𝑧𝑦subscript𝜔0\displaystyle=\frac{1}{\sqrt{h}}\int_{0}^{h}u^{n}(h,y,z)dz,\ \ y\in\omega_{0}, (4.11)
vjn(h,z)superscriptsubscript𝑣𝑗𝑛𝑧\displaystyle v_{j}^{n}(h,z) =1h3/2|ωj|ωjhujn(h,y,z)𝑑y,y(0,lj),j=1,,J,formulae-sequenceabsent1superscript32subscript𝜔𝑗subscriptsuperscriptsubscript𝜔𝑗superscriptsubscript𝑢𝑗𝑛𝑦𝑧differential-d𝑦formulae-sequence𝑦0subscript𝑙𝑗𝑗1𝐽\displaystyle=\frac{1}{h^{3/2}|\omega_{j}|}\int_{\omega_{j}^{h}}u_{j}^{n}(h,y,z)dy,\ \ y\in(0,l_{j}),\ j=1,...,J, (4.12)

and write

ω0|v0n(h,y)|2𝑑ysubscriptsubscript𝜔0superscriptsuperscriptsubscript𝑣0𝑛𝑦2differential-d𝑦\displaystyle\int_{\omega_{0}}|v_{0}^{n}(h,y)|^{2}dy =1hω0|0hun(h,y,z)𝑑z|2𝑑yω00h|un(h,x)|2𝑑xabsent1subscriptsubscript𝜔0superscriptsuperscriptsubscript0superscript𝑢𝑛𝑦𝑧differential-d𝑧2differential-d𝑦subscriptsubscript𝜔0superscriptsubscript0superscriptsuperscript𝑢𝑛𝑥2differential-d𝑥\displaystyle=\frac{1}{h}\int_{\omega_{0}}\left|\int_{0}^{h}u^{n}(h,y,z)dz\right|^{2}dy\leq\int_{\omega_{0}}\int_{0}^{h}\left|u^{n}(h,x)\right|^{2}dx (4.13)
b(un,un;Ω0(h)),absent𝑏superscript𝑢𝑛superscript𝑢𝑛subscriptΩ0\displaystyle\leq b(u^{n},u^{n};\Omega_{0}(h)),
ρj|ωj|0lj|vjn(h,z)|2𝑑zsubscript𝜌𝑗subscript𝜔𝑗superscriptsubscript0subscript𝑙𝑗superscriptsuperscriptsubscript𝑣𝑗𝑛𝑧2differential-d𝑧\displaystyle\rho_{j}|\omega_{j}|\int_{0}^{l_{j}}|v_{j}^{n}(h,z)|^{2}dz =ρj|ωj|11h30lj|ωjhujn(h,y,z)𝑑y|2𝑑zabsentsubscript𝜌𝑗superscriptsubscript𝜔𝑗11superscript3superscriptsubscript0subscript𝑙𝑗superscriptsubscriptsuperscriptsubscript𝜔𝑗superscriptsubscript𝑢𝑗𝑛𝑦𝑧differential-d𝑦2differential-d𝑧\displaystyle=\rho_{j}|\omega_{j}|^{-1}\frac{1}{h^{3}}\int_{0}^{l_{j}}\left|\int_{\omega_{j}^{h}}u_{j}^{n}(h,y,z)dy\right|^{2}dz
ρjh|ωjh|h2|ωj|Ωjh(h)|un(h,x)|2𝑑xb(un,un;Ωj(h)).absentsubscript𝜌𝑗superscriptsubscript𝜔𝑗superscript2subscript𝜔𝑗subscriptsuperscriptsubscriptΩ𝑗superscriptsuperscript𝑢𝑛𝑥2differential-d𝑥𝑏superscript𝑢𝑛superscript𝑢𝑛subscriptΩ𝑗\displaystyle\leq\frac{\rho_{j}}{h}\frac{|\omega_{j}^{h}|}{h^{2}|\omega_{j}|}\int_{\Omega_{j}^{h}(h)}|u^{n}(h,x)|^{2}dx\leq b(u^{n},u^{n};\Omega_{j}(h)).

Here, we used formulas (1.17) and (1.13), (1.14) while taking the relation 1h1ρj1superscript1subscript𝜌𝑗1\leq h^{-1}\rho_{j} on Ω0(h)Ωj(h)subscriptΩ0subscriptΩ𝑗\Omega_{0}(h)\cap\Omega_{j}(h) into account. Hence, the vector function vn=(v0n,v1n,,vJn)superscript𝑣𝑛superscriptsubscript𝑣0𝑛superscriptsubscript𝑣1𝑛superscriptsubscript𝑣𝐽𝑛v^{n}=(v_{0}^{n},v_{1}^{n},...,v_{J}^{n}) satisfies the estimate

||vn(h,);𝔏||1.||v^{n}(h,\cdot);\mathfrak{L}||\leq 1. (4.14)

A similar calculation gives us the formula

||v0n;L2(ω0)||2+j||zvjn;L2(Ij)||2\displaystyle||\nabla v_{0}^{n};L^{2}(\omega_{0})||^{2}+{\textstyle\sum\nolimits_{j}}||\partial_{z}v_{j}^{n};L^{2}(I_{j})||^{2} c(||yun;L2(Ω0(h))||2+j||zun;L2(Ωj(h))||2)\displaystyle\leq c(||\nabla_{y}u^{n};L^{2}(\Omega_{0}(h))||^{2}+{\textstyle\sum\nolimits_{j}}||\partial_{z}u^{n};L^{2}(\Omega_{j}(h))||^{2})
ca(un,un;Ξ(h))=cλ(h)b(un,un;Ξ(h))Cn.absent𝑐𝑎superscript𝑢𝑛superscript𝑢𝑛Ξ𝑐𝜆𝑏superscript𝑢𝑛superscript𝑢𝑛Ξsubscript𝐶𝑛\displaystyle\leq ca(u^{n},u^{n};\Xi(h))=c\lambda(h)b(u^{n},u^{n};\Xi(h))\leq C_{n}.

Moreover, the Poincaré inequalities in (0,h)0(0,h) and ωjhsuperscriptsubscript𝜔𝑗\omega_{j}^{h} show that the functions

u0n(h,x)superscriptsubscript𝑢0perpendicular-to𝑛absent𝑥\displaystyle u_{0}^{n\perp}(h,x) =un(h,x)h1/2v0n(h,y) in Ω0(h),absentsuperscript𝑢𝑛𝑥superscript12superscriptsubscript𝑣0𝑛𝑦 in subscriptΩ0\displaystyle=u^{n}(h,x)-h^{-1/2}v_{0}^{n}(h,y)\text{\ \ in }\Omega_{0}(h),
ujn(h,x)superscriptsubscript𝑢𝑗perpendicular-to𝑛absent𝑥\displaystyle u_{j}^{n\perp}(h,x) =ujn(h,x)h1/2vjn(h,y) in Ωj(h),absentsuperscriptsubscript𝑢𝑗𝑛𝑥superscript12superscriptsubscript𝑣𝑗𝑛𝑦 in subscriptΩ𝑗\displaystyle=u_{j}^{n}(h,x)-h^{-1/2}v_{j}^{n}(h,y)\text{\ \ in }\Omega_{j}(h),

which are of mean zero in z(0,h)𝑧0z\in(0,h) and yωjh𝑦superscriptsubscript𝜔𝑗y\in\omega_{j}^{h}, respectively, enjoy the relations

||u0n;L2(Ω0(h))||2\displaystyle||u_{0}^{n\perp};L^{2}(\Omega_{0}(h))||^{2} c0h2||z(unh1/2v0n);L2(Ω0(h))||2\displaystyle\leq c_{0}h^{2}||\partial_{z}(u^{n}-h^{-1/2}v_{0}^{n});L^{2}(\Omega_{0}(h))||^{2} (4.15)
=c0h2||zun;L2(Ω0(h))||2c0a(un,un;Ω0(h)),\displaystyle=c_{0}h^{2}||\partial_{z}u^{n};L^{2}(\Omega_{0}(h))||^{2}\leq c_{0}a(u^{n},u^{n};\Omega_{0}(h)),
||ujn;L2(Ωj(h))||2\displaystyle||u_{j}^{n\perp};L^{2}(\Omega_{j}(h))||^{2} cjh2||y(ujnh1/2vjn);L2(Ωj(h))||2\displaystyle\leq c_{j}h^{2}||\nabla_{y}(u_{j}^{n}-h^{-1/2}v_{j}^{n});L^{2}(\Omega_{j}(h))||^{2}
=cjh2||yun;L2(Ωj(h))||2Cjh3a(un,un;Ωj(h)).\displaystyle=c_{j}h^{2}||\nabla_{y}u^{n};L^{2}(\Omega_{j}(h))||^{2}\leq C_{j}h^{3}a(u^{n},u^{n};\Omega_{j}(h)).

Hence, we obtain

11\displaystyle 1 =||h1/2v0nu0n;L2(Ω(h))||2+h1jρj||h1/2vjnujn;L2(Ωj(h))||2\displaystyle=||h^{-1/2}v_{0}^{n}-u_{0}^{n\perp};L^{2}(\Omega_{\bullet}(h))||^{2}+h^{-1}{\textstyle\sum\nolimits_{j}}\rho_{j}||h^{-1/2}v_{j}^{n}-u_{j}^{n\perp};L^{2}(\Omega_{j}(h))||^{2} (4.16)
(1+h)(||v0n;L2(ω(h))||2+jρj|ωj|||vjn;L2(Ij)||2)\displaystyle\leq(1+h)(||v_{0}^{n};L^{2}(\omega_{\bullet}(h))||^{2}+{\textstyle\sum\nolimits_{j}}\rho_{j}|\omega_{j}|~{}||v_{j}^{n};L^{2}(I_{j})||^{2})
+(1+h1)(||u0n;L2(Ω(h))||2+h1jρj||ujn;L2(Ωj(h))||2).\displaystyle+(1+h^{-1})(||u_{0}^{n\perp};L^{2}(\Omega_{\bullet}(h))||^{2}+h^{-1}{\textstyle\sum\nolimits_{j}}\rho_{j}||u_{j}^{n\perp};L^{2}(\Omega_{j}(h))||^{2}).

To estimate the norm ||v0n;L2(ω(h))||2||v_{0}^{n};L^{2}(\omega_{\bullet}(h))||^{2}, we apply the weighted inequality in [4, Thm. 9]

(1+|lnh|)2||r1(1+|lnr|)1u;L2(Ω0(h))||2+h1j||(ljz)1uj;L2(Ωj(h))||2\displaystyle(1+|\ln h|)^{-2}||r^{-1}(1+|\ln r|)^{-1}u;L^{2}(\Omega_{0}(h))||^{2}+h^{-1}{\textstyle\sum\nolimits_{j}}||(l_{j}-z)^{-1}u_{j};L^{2}(\Omega_{j}(h))||^{2} (4.17)
cΞa(u,u;Ξ(h))absentsubscript𝑐Ξ𝑎𝑢𝑢Ξ\displaystyle\leq c_{\Xi}a(u,u;\Xi(h))

where r=min{r1,,rJ}𝑟subscript𝑟1subscript𝑟𝐽r=\min\{r_{1},...,r_{J}\} and cΞsubscript𝑐Ξc_{\Xi} is independent of h(0,h0]0subscript0h\in(0,h_{0}] and uH01(Ξ(h);Γ(h))𝑢superscriptsubscript𝐻01ΞΓu\in H_{0}^{1}(\Xi(h);\Gamma(h)). Since rj(1+|lnrj|)cjh(1+|lnh|)subscript𝑟𝑗1subscript𝑟𝑗subscript𝑐𝑗1r_{j}(1+|\ln r_{j}|)\leq c_{j}h(1+|\ln h|) in the socket θjh=ωjh×(0,h)superscriptsubscript𝜃𝑗superscriptsubscript𝜔𝑗0\theta_{j}^{h}=\omega_{j}^{h}\times(0,h), we recall (1.16), (4.10) and conclude that

||v0\displaystyle||v_{0} ;nL2(ωjh)||2||un;L2(θjh)||2ch2(1+|lnh|)4||rj1(1+|lnrj|)1un;L2(θjh)||2{}^{n};L^{2}(\omega_{j}^{h})||^{2}\leq||u^{n};L^{2}(\theta_{j}^{h})||^{2}\leq ch^{2}(1+|\ln h|)^{4}||r_{j}^{-1}(1+|\ln r_{j}|)^{-1}u^{n};L^{2}(\theta_{j}^{h})||^{2} (4.18)
ch2(1+|lnh|)4a(un,un;Ξ(h))=ch2(1+|lnh|)4λn(h)b(un,un;Ξ(h))Cnh2(1+|lnh|)4.absent𝑐superscript2superscript14𝑎superscript𝑢𝑛superscript𝑢𝑛Ξ𝑐superscript2superscript14superscript𝜆𝑛𝑏superscript𝑢𝑛superscript𝑢𝑛Ξsubscript𝐶𝑛superscript2superscript14\displaystyle\leq ch^{2}(1+|\ln h|)^{4}a(u^{n},u^{n};\Xi(h))=ch^{2}(1+|\ln h|)^{4}\lambda^{n}(h)b(u^{n},u^{n};\Xi(h))\leq C_{n}h^{2}(1+|\ln h|)^{4}.

The estimates (4.10) and (4.13), (4.14) provide the following convergence along an infinitesimal sequence {hk}ksubscriptsubscript𝑘𝑘\{h_{k}\}_{k\in\mathbb{N}}:

λn(h)superscript𝜆𝑛\displaystyle\lambda^{n}(h) λ0,absentsuperscript𝜆0\displaystyle\rightarrow\lambda^{0}, (4.19)
v0n(h,)superscriptsubscript𝑣0𝑛\displaystyle v_{0}^{n}(h,\cdot) v0n0 weakly in H1(ω0) and strongly in L2(ω0),absentsuperscriptsubscript𝑣0𝑛0 weakly in superscript𝐻1subscript𝜔0 and strongly in superscript𝐿2subscript𝜔0\displaystyle\rightharpoondown v_{0}^{n0}\text{ weakly in }H^{1}(\omega_{0})\text{ and strongly in }L^{2}(\omega_{0}),
vjn(h,)superscriptsubscript𝑣𝑗𝑛\displaystyle v_{j}^{n}(h,\cdot) vjn0 weakly in H1(Ij) and strongly in L2(Ij).absentsuperscriptsubscript𝑣𝑗𝑛0 weakly in superscript𝐻1subscript𝐼𝑗 and strongly in superscript𝐿2subscript𝐼𝑗\displaystyle\rightharpoondown v_{j}^{n0}\text{ weakly in }H^{1}(I_{j})\text{ and strongly in }L^{2}(I_{j}).

We compose a test function w𝑤w from components w0Cc(ω¯0𝒫)subscript𝑤0superscriptsubscript𝐶𝑐subscript¯𝜔0𝒫w_{0}\in C_{c}^{\infty}(\overline{\omega}_{0}\setminus\mathcal{P}) and wjCc(Ij),subscript𝑤𝑗superscriptsubscript𝐶𝑐subscript𝐼𝑗w_{j}\in C_{c}^{\infty}(I_{j}), j=1,,J𝑗1𝐽j=1,...,J. Then according to (1.17) and (4.11), (4.12), we transform the integral identity into the formula

h(y\displaystyle\sqrt{h}(\nabla_{y} v0n,yw0)ω0+hjγj|ωj|(zvjn,zwj)Ij=(yun,yw0)Ω(h)+h1jγj(zujn,zwj)Ωj(h)\displaystyle v_{0}^{n},\nabla_{y}w_{0})_{\omega_{0}}+\sqrt{h}{\textstyle\sum\nolimits_{j}}\gamma_{j}|\omega_{j}|(\partial_{z}v_{j}^{n},\partial_{z}w_{j})_{I_{j}}=(\nabla_{y}u^{n},\nabla_{y}w_{0})_{\Omega_{\bullet}(h)}+h^{-1}{\textstyle\sum\nolimits_{j}}\gamma_{j}(\partial_{z}u_{j}^{n},\partial_{z}w_{j})_{\Omega_{j}(h)}
=λ(h)((un,w0)Ω(h)+h1jρj(ujn,wj)Ωj(h))=hλ(h)((v0n,w0)ω0+jρj|ωj|(vjn,wj)Ij).absent𝜆subscriptsuperscript𝑢𝑛subscript𝑤0subscriptΩsuperscript1subscript𝑗subscript𝜌𝑗subscriptsuperscriptsubscript𝑢𝑗𝑛subscript𝑤𝑗subscriptΩ𝑗𝜆subscriptsuperscriptsubscript𝑣0𝑛subscript𝑤0subscript𝜔0subscript𝑗subscript𝜌𝑗subscript𝜔𝑗subscriptsuperscriptsubscript𝑣𝑗𝑛subscript𝑤𝑗subscript𝐼𝑗\displaystyle=\lambda(h)\left((u^{n},w_{0})_{\Omega_{\bullet}(h)}+h^{-1}{\textstyle\sum\nolimits_{j}}\rho_{j}(u_{j}^{n},w_{j})_{\Omega_{j}(h)}\right)=\sqrt{h}\lambda(h)\left((v_{0}^{n},w_{0})_{\omega_{0}}+{\textstyle\sum\nolimits_{j}}\rho_{j}|\omega_{j}|(v_{j}^{n},w_{j})_{I_{j}}\right).

We multiply this with h1/2superscript12h^{-1/2} and perform the limit passage along the sequence {hk}ksubscriptsubscript𝑘𝑘\{h_{k}\}_{k\in\mathbb{N}} to obtain

(yv0n0,yw0)ω0+jγj|ωj|(zvjn0,zwj)Ij=λ0((v0n0,w0)ω0+jρj|ωj|(vjn0,wj)Ij).subscriptsubscript𝑦superscriptsubscript𝑣0𝑛0subscript𝑦subscript𝑤0subscript𝜔0subscript𝑗subscript𝛾𝑗subscript𝜔𝑗subscriptsubscript𝑧superscriptsubscript𝑣𝑗𝑛0subscript𝑧subscript𝑤𝑗subscript𝐼𝑗superscript𝜆0subscriptsuperscriptsubscript𝑣0𝑛0subscript𝑤0subscript𝜔0subscript𝑗subscript𝜌𝑗subscript𝜔𝑗subscriptsuperscriptsubscript𝑣𝑗𝑛0subscript𝑤𝑗subscript𝐼𝑗(\nabla_{y}v_{0}^{n0},\nabla_{y}w_{0})_{\omega_{0}}+{\textstyle\sum\nolimits_{j}}\gamma_{j}|\omega_{j}|(\partial_{z}v_{j}^{n0},\partial_{z}w_{j})_{I_{j}}=\lambda^{0}\left((v_{0}^{n0},w_{0})_{\omega_{0}}+{\textstyle\sum\nolimits_{j}}\rho_{j}|\omega_{j}|(v_{j}^{n0},w_{j})_{I_{j}}\right). (4.20)

Thanks to [15, §9 Ch. 2], the weak solution v0n0H1(ω0)superscriptsubscript𝑣0𝑛0superscript𝐻1subscript𝜔0v_{0}^{n0}\in H^{1}(\omega_{0}) of the variational problem (4.20) where wj=0,subscript𝑤𝑗0w_{j}=0, j=1,,J𝑗1𝐽j=1,...,J, falls into H2(ω0)superscript𝐻2subscript𝜔0H^{2}(\omega_{0}) and satisfies the Neumann problem (1.20), (1.22) with μ=λ0𝜇superscript𝜆0\mu=\lambda^{0}. We emphasize that Cc(ω¯0𝒫)superscriptsubscript𝐶𝑐subscript¯𝜔0𝒫C_{c}^{\infty}(\overline{\omega}_{0}\setminus\mathcal{P}) is dense in H1(ω0)superscript𝐻1subscript𝜔0H^{1}(\omega_{0}) so that any test function wH1(ω0)𝑤superscript𝐻1subscript𝜔0w\in H^{1}(\omega_{0}) is available. At the same time, wjCc(Ij)subscript𝑤𝑗superscriptsubscript𝐶𝑐subscript𝐼𝑗w_{j}\in C_{c}^{\infty}(I_{j}) vanishes near the points z=lj𝑧subscript𝑙𝑗z=l_{j} and z=0𝑧0z=0. Hence, we may conclude that vjn0H2(Ij)superscriptsubscript𝑣𝑗𝑛0superscript𝐻2subscript𝐼𝑗v_{j}^{n0}\in H^{2}(I_{j}) and the differential equation (1.21) with μ=λ0𝜇superscript𝜆0\mu=\lambda^{0}. However, the boundary conditions

vjn0(lj)=0,γj|ωj|zvjn0(0)=0formulae-sequencesuperscriptsubscript𝑣𝑗𝑛0subscript𝑙𝑗0subscript𝛾𝑗subscript𝜔𝑗subscript𝑧superscriptsubscript𝑣𝑗𝑛000v_{j}^{n0}(l_{j})=0,\ -\gamma_{j}|\omega_{j}|\partial_{z}v_{j}^{n0}(0)=0 (4.21)

still must be derived. The Dirichlet condition in (4.21) is inherited from the conditions vjn(h,lj)=0superscriptsubscript𝑣𝑗𝑛subscript𝑙𝑗0v_{j}^{n}(h,l_{j})=0 and ujn(h,y,lj)=0,superscriptsubscript𝑢𝑗𝑛𝑦subscript𝑙𝑗0u_{j}^{n}(h,y,l_{j})=0, yωjh𝑦superscriptsubscript𝜔𝑗y\in\omega_{j}^{h}. To conclude with the Neumann condition, we observe that the inequality (4.18) allows us to repeat the above transformations with the ”very special” test vector function

w0j(y)=χj(y),wkj(z)=Xk(z)δk,j,j,k=1,,J,formulae-sequencesuperscriptsubscript𝑤0𝑗𝑦subscript𝜒𝑗𝑦formulae-sequencesuperscriptsubscript𝑤𝑘𝑗𝑧subscript𝑋𝑘𝑧subscript𝛿𝑘𝑗𝑗𝑘1𝐽w_{0}^{j}(y)=\chi_{j}(y),\ \ w_{k}^{j}(z)=X_{k}(z)\delta_{k,j},\ \ j,k=1,...,J, (4.22)

where χjsubscript𝜒𝑗\chi_{j} and Xksubscript𝑋𝑘X_{k} are taken from (2.5) and (3.19). As a result, the obtained information on v0n0superscriptsubscript𝑣0𝑛0v_{0}^{n0} and vjn0superscriptsubscript𝑣𝑗𝑛0v_{j}^{n0} reduces the integral identity (4.21) with (4.22) to the formula

0=((Δy+λ0)v0n0,w0j)ω0γj|ωj|(((z2+λ0)vjn0,zwjj)Ij+zvjn0(0)wjj(0))=γj|ωj|zvjn0(0).0subscriptsubscriptΔ𝑦superscript𝜆0superscriptsubscript𝑣0𝑛0superscriptsubscript𝑤0𝑗subscript𝜔0subscript𝛾𝑗subscript𝜔𝑗subscriptsuperscriptsubscript𝑧2superscript𝜆0superscriptsubscript𝑣𝑗𝑛0subscript𝑧superscriptsubscript𝑤𝑗𝑗subscript𝐼𝑗subscript𝑧superscriptsubscript𝑣𝑗𝑛00superscriptsubscript𝑤𝑗𝑗0subscript𝛾𝑗subscript𝜔𝑗subscript𝑧superscriptsubscript𝑣𝑗𝑛000=-((\Delta_{y}+\lambda^{0})v_{0}^{n0},w_{0}^{j})_{\omega_{0}}-\gamma_{j}|\omega_{j}|(((\partial_{z}^{2}+\lambda^{0})v_{j}^{n0},\partial_{z}w_{j}^{j})_{I_{j}}+\partial_{z}v_{j}^{n0}(0)w_{j}^{j}(0))=-\gamma_{j}|\omega_{j}|\partial_{z}v_{j}^{n0}(0).

We are in position to formulate the convergence theorem.

Theorem 14

The limits λ0superscript𝜆0\lambda^{0} and vn0=(v0n0,v1n0,,vJn0)superscript𝑣𝑛0superscriptsubscript𝑣0𝑛0superscriptsubscript𝑣1𝑛0superscriptsubscript𝑣𝐽𝑛0v^{n0}=(v_{0}^{n0},v_{1}^{n0},...,v_{J}^{n0}) in (4.19) are an eigenvalue and the corresponding eigenvector normed by (4.1) of the problems (1.20), (1.22) and (1.21), (4.21).

4.3 An abstract formulation of the original problem

In the Hilbert space Hh=H01(Ξ(h);Γ(h))superscript𝐻superscriptsubscript𝐻01ΞΓH^{h}=H_{0}^{1}(\Xi(h);\Gamma(h)) we introduce the scalar product

(uh,vh)h=a(uh,vh;Ξ(h))+b(uh,vh;Ξ(h))subscriptsuperscript𝑢superscript𝑣𝑎superscript𝑢superscript𝑣Ξ𝑏superscript𝑢superscript𝑣Ξ(u^{h},v^{h})_{h}=a(u^{h},v^{h};\Xi(h))+b(u^{h},v^{h};\Xi(h)) (4.23)

and positive, symmetric and continuous, therefore, self-adjoint operator Thsuperscript𝑇T^{h},

(Thuh,vh)h=b(uh,vh;Ξ(h))uh,vhHh.formulae-sequencesubscriptsuperscript𝑇superscript𝑢superscript𝑣𝑏superscript𝑢superscript𝑣Ξfor-allsuperscript𝑢superscript𝑣superscript𝐻(T^{h}u^{h},v^{h})_{h}=b(u^{h},v^{h};\Xi(h))\ \ \ \ \forall u^{h},v^{h}\in H^{h}. (4.24)

Bilinear form on the right-hand side of (4.23) are defined in (1.17). Comparing (1.16) with (4.23), (4.24), we see that the variational formulation of the problem (1.6)-(1.12) in Ξ(h)Ξ\Xi(h) is equivalent to the abstract equation

Thuh=τ(h)uh in Hhsuperscript𝑇superscript𝑢𝜏superscript𝑢 in superscript𝐻T^{h}u^{h}=\tau(h)u^{h}\text{ \ in }H^{h} (4.25)

with the new spectral parameter

τ(h)=(1+λ(h))1.𝜏superscript1𝜆1\tau(h)=(1+\lambda(h))^{-1}. (4.26)

The operator Thsuperscript𝑇T^{h} is compact and, hence, the essential spectrum of Thsuperscript𝑇T^{h} consists of the only point τ=0𝜏0\tau=0, see, e.g., [2, Thm. 10.1.5], while the discrete spectrum composes the positive monotone infinitesimal sequence

1>τ1(h)>τ2(h)τn(h).+01>\tau^{1}(h)>\tau^{2}(h)\geq...\geq\tau^{n}(h)\geq....\rightarrow+0 (4.27)

obtained from (1.18) according to formula (4.26).

The following assertion is known as the lemma on ”near eigenvalues and eigenvectors” [34] following directly from the spectral decomposition of resolvent, see, e.g., [2, Ch. 6].

Lemma 15

Let 𝒰hHhsuperscript𝒰superscript𝐻\mathcal{U}^{h}\in H^{h} and th+superscript𝑡subscriptt^{h}\in\mathbb{R}_{+} satisfy

||𝒰h;Hh||=1,||Th𝒰hth𝒰h;Hh||:=δh(0,th).||\mathcal{U}^{h};H^{h}||=1,\ ||T^{h}\mathcal{U}^{h}-t^{h}\mathcal{U}^{h};H^{h}||:=\delta^{h}\in(0,t^{h}). (4.28)

Then there exists an eigenvalue τnhsuperscriptsubscript𝜏𝑛\tau_{n}^{h} of the operator Thsuperscript𝑇T^{h} such that

|thτnh|δh.superscript𝑡superscriptsubscript𝜏𝑛superscript𝛿|t^{h}-\tau_{n}^{h}|\leq\delta^{h}. (4.29)

Moreover, for any δh(δh,th)superscriptsubscript𝛿superscript𝛿superscript𝑡\delta_{\ast}^{h}\in(\delta^{h},t^{h}), one finds coefficients ckhsuperscriptsubscript𝑐𝑘c_{k}^{h} to fulfil the relations

𝒰hk=NhNh+Xh1ckhu(k)h;Hh2δhδh,k=NhNh+Xh1|ckh|2=1\left\|\mathcal{U}^{h}-\sum\nolimits_{k=N^{h}}^{N^{h}+X^{h}-1}c_{k}^{h}u_{(k)}^{h};H^{h}\right\|\leq 2\frac{\delta^{h}}{\delta_{\ast}^{h}},\ \ \ \ \sum\nolimits_{k=N^{h}}^{N^{h}+X^{h}-1}|c_{k}^{h}|^{2}=1 (4.30)

where τNhh,,τNh+Xh1hsuperscriptsubscript𝜏superscript𝑁superscriptsubscript𝜏superscript𝑁superscript𝑋1\tau_{N^{h}}^{h},...,\tau_{N^{h}+X^{h}-1}^{h}are all eigenvalues in the segment [thδh,th+δh]superscript𝑡superscriptsubscript𝛿superscript𝑡superscriptsubscript𝛿[t^{h}-\delta_{\ast}^{h},t^{h}+\delta_{\ast}^{h}] and u(Nh)h,,u(Nh+Xh1)hsuperscriptsubscript𝑢superscript𝑁superscriptsubscript𝑢superscript𝑁superscript𝑋1u_{(N^{h})}^{h},...,u_{(N^{h}+X^{h}-1)}^{h} are the corresponding eigenvectors subject to the normalization and orthogonality conditions

(u(p)h,u(q)h)h=δp,q.subscriptsuperscriptsubscript𝑢𝑝superscriptsubscript𝑢𝑞subscript𝛿𝑝𝑞(u_{(p)}^{h},u_{(q)}^{h})_{h}=\delta_{p,q}. (4.31)

4.4 Detecting eigenvalues with prescribed asymptotic form

Let μph+superscriptsubscript𝜇𝑝subscript\mu_{p}^{h}\in\mathbb{R}_{+} and v(p)h=(v(p)0h,v(p)1h,,v(p)Jh)superscriptsubscript𝑣𝑝superscriptsubscript𝑣𝑝0superscriptsubscript𝑣𝑝1superscriptsubscript𝑣𝑝𝐽subscriptv_{(p)}^{h}=(v_{(p)0}^{h},v_{(p)1}^{h},...,v_{(p)J}^{h})\in\mathfrak{H}_{-} be an eigenvalue of the equation (3.15) and the corresponding eigenvector enjoing the normalization and orthogonality conditions

(v(p)h,v(q)h)𝔏=δp,qsubscriptsuperscriptsubscript𝑣𝑝superscriptsubscript𝑣𝑞𝔏subscript𝛿𝑝𝑞(v_{(p)}^{h},v_{(q)}^{h})_{\mathfrak{L}}=\delta_{p,q} (4.32)

where (,)𝔏(\ ,\ )_{\mathfrak{L}} denotes the scalar product in the Lebesgue space 𝔏𝔏\mathfrak{L} induced by the norm (2.14). In Lemma 15 we set

tph=(1+μph)1superscriptsubscript𝑡𝑝superscript1superscriptsubscript𝜇𝑝1t_{p}^{h}=(1+\mu_{p}^{h})^{-1} (4.33)

and build an asymptotic approximation 𝒰(p)hsuperscriptsubscript𝒰𝑝\mathcal{U}_{(p)}^{h} of an eigenfunction of the problem (1.6)-(1.12) in the junction (1.4). To mimic the method of matched asymptotic expansions, we use asymptotic structures with ”overlapping” cut-off functions, see [16, Ch.2], [22, 18] etc., namely in addition to the functions χjsubscript𝜒𝑗\chi_{j} in (2.5) and Xjsubscript𝑋𝑗X_{j} in (3.19) we introduce

𝒳h(y)superscript𝒳𝑦\displaystyle\mathcal{X}^{h}(y) =0 for rjRh,j=1,,J,𝒳h(y)=1 for min{r1,,rJ}2Rh,formulae-sequenceabsent0 for subscript𝑟𝑗𝑅formulae-sequence𝑗1𝐽superscript𝒳𝑦1 for subscript𝑟1subscript𝑟𝐽2𝑅\displaystyle=0\text{ for }r_{j}\leq Rh,\ j=1,...,J,\ \ \ \ \mathcal{X}^{h}(y)=1\text{ for }\min\{r_{1},...,r_{J}\}\geq 2Rh, (4.34)
Xh(z)superscript𝑋𝑧\displaystyle X^{h}(z) =0 for z2h,Xh(z)=1 for z3h.formulae-sequenceabsent0 for 𝑧2superscript𝑋𝑧1 for 𝑧3\displaystyle=0\text{ for }z\leq 2h,\ \ \ \ X^{h}(z)=1\text{ for }z\geq 3h.

Radius R𝑅R is chosen such that 𝒳h(y)=0superscript𝒳𝑦0\mathcal{X}^{h}(y)=0 on ωjh.superscriptsubscript𝜔𝑗\omega_{j}^{h}.

The functions 𝒰(p)hsuperscriptsubscript𝒰𝑝\mathcal{U}_{(p)}^{h} and 𝒱(p)hsuperscriptsubscript𝒱𝑝\mathcal{V}_{(p)}^{h} are determined by formulas

𝒰(p)hsuperscriptsubscript𝒰𝑝\displaystyle\mathcal{U}_{(p)}^{h} =||𝒱(p)h;Hh||1𝒱(p)h,\displaystyle=||\mathcal{V}_{(p)}^{h};H^{h}||^{-1}\mathcal{V}_{(p)}^{h}, (4.35)
𝒱(p)0h(x)superscriptsubscript𝒱𝑝0𝑥\displaystyle\mathcal{V}_{(p)0}^{h}(x) =𝒳h(y)v(p)0h(y)+jχj(y)(cj0+cj1(𝐖j0(ξj)+h𝐰^j(ξj))\displaystyle=\mathcal{X}^{h}(y)v_{(p)0}^{h}(y)+{\textstyle\sum\nolimits_{j}}\chi_{j}(y)(c_{j}^{0}+c_{j}^{1}(\mathbf{W}_{j}^{0}(\xi^{j})+h\widehat{\mathbf{w}}_{j}(\xi^{j})) (4.36)
j𝒳h(y)χj(y)(cj0+cj112π(ln1|ηj|+lnclog(ωj)),\displaystyle-{\textstyle\sum\nolimits_{j}}\mathcal{X}^{h}(y)\chi_{j}(y)\left(c_{j}^{0}+c_{j}^{1}\frac{1}{2\pi}(\ln\frac{1}{|\eta^{j}|}+\ln c_{\log}(\omega_{j})\right),
𝒱(p)jh(x)superscriptsubscript𝒱𝑝𝑗𝑥\displaystyle\mathcal{V}_{(p)j}^{h}(x) =Xh(z)v(p)jh(z)+Xj(z)(cj0+hcj1𝐖j(ξj))Xh(z)Xj(z)(cj0+hcj1γj1|ωj|1ζj)absentsuperscript𝑋𝑧superscriptsubscript𝑣𝑝𝑗𝑧subscript𝑋𝑗𝑧superscriptsubscript𝑐𝑗0superscriptsubscript𝑐𝑗1subscript𝐖𝑗superscript𝜉𝑗superscript𝑋𝑧subscript𝑋𝑗𝑧superscriptsubscript𝑐𝑗0superscriptsubscript𝑐𝑗1superscriptsubscript𝛾𝑗1superscriptsubscript𝜔𝑗1superscript𝜁𝑗\displaystyle=X^{h}(z)v_{(p)j}^{h}(z)+X_{j}(z)(c_{j}^{0}+hc_{j}^{1}\mathbf{W}_{j}(\xi^{j}))-X^{h}(z)X_{j}(z)(c_{j}^{0}+hc_{j}^{1}\gamma_{j}^{-1}|\omega_{j}|^{-1}\zeta^{j}) (4.37)

where ingredients are taken from (3.9), (3.10) and 𝐰^jH1(Λj)subscript^𝐰𝑗superscript𝐻1subscriptΛ𝑗\widehat{\mathbf{w}}_{j}\in H^{1}(\Lambda_{j}) is a function with compact support such that

𝐰^j(ξj)=𝐖j(ξj),ξjωj×(0,1).formulae-sequencesubscript^𝐰𝑗superscript𝜉𝑗subscript𝐖𝑗superscript𝜉𝑗superscript𝜉𝑗subscript𝜔𝑗01\widehat{\mathbf{w}}_{j}(\xi^{j})=\mathbf{W}_{j}(\xi^{j}),\ \ \xi^{j}\in\partial\omega_{j}\times(0,1). (4.38)

The latter condition and the cut-off functions 𝒳h,Xhsuperscript𝒳superscript𝑋\mathcal{X}^{h},X^{h} in (4.36), (4.37) assure that 𝒱(p)0hsuperscriptsubscript𝒱𝑝0\mathcal{V}_{(p)0}^{h} and 𝒱(p)jhsuperscriptsubscript𝒱𝑝𝑗\mathcal{V}_{(p)j}^{h} coincide with each other on υj(h)subscript𝜐𝑗\upsilon_{j}(h), cf. (1.11), and the composite function 𝒱(p)hsuperscriptsubscript𝒱𝑝\mathcal{V}_{(p)}^{h} falls into H01(Ξ(h);Γ(h)).superscriptsubscript𝐻01ΞΓH_{0}^{1}(\Xi(h);\Gamma(h)).

First of all, we compute the scalar products (𝒱(p)h,𝒱(q)h)hsubscriptsuperscriptsubscript𝒱𝑝superscriptsubscript𝒱𝑞(\mathcal{V}_{(p)}^{h},\mathcal{V}_{(q)}^{h})_{h}. To this end, we observe that, according to (3.12), we have

j(|b(p)jh|+|zv(p)jh(0)|)c|lnh|1j(|v^(p)0h(Pj)|+|v(p)jh(0)|).subscript𝑗superscriptsubscript𝑏𝑝𝑗subscript𝑧superscriptsubscript𝑣𝑝𝑗0𝑐superscript1subscript𝑗superscriptsubscript^𝑣𝑝0superscript𝑃𝑗superscriptsubscript𝑣𝑝𝑗0{\textstyle\sum\nolimits_{j}}(|b_{(p)j}^{h}|+|\partial_{z}v_{(p)j}^{h}(0)|)\leq c|\ln h|^{-1}{\textstyle\sum\nolimits_{j}}(|\widehat{v}_{(p)0}^{h}(P^{j})|+|v_{(p)j}^{h}(0)|). (4.39)

The estimate (1.9) applied in the problem (1.20), (1.21) shows that

||v^(p)0h;H1(ω0)||+j|v^(p)0h(Pj)|cμp||v(p)0h;L2(ω0)||Cp.||\widehat{v}_{(p)0}^{h};H^{1}(\omega_{0})||+{\textstyle\sum\nolimits_{j}}|\widehat{v}_{(p)0}^{h}(P^{j})|\leq c\mu_{p}||v_{(p)0}^{h};L^{2}(\omega_{0})||\leq C_{p}. (4.40)

Moreover, a solution of (1.21), (1.23) satisfies

||v(p)j;H2(Ij)||c(μp||v(p)j;L2(Ij)||+|zv(p)j(0)|)||v_{(p)j};H^{2}(I_{j})||\leq c(\mu_{p}||v_{(p)j};L^{2}(I_{j})||+|\partial_{z}v_{(p)j}(0)|) (4.41)

while the last bound is due to the estimate (4.40) and the small factor |lnh|1superscript1|\ln h|^{-1} on the right of (4.39).

Recalling that cj0=v^(p)0h(Pj)superscriptsubscript𝑐𝑗0superscriptsubscript^𝑣𝑝0superscript𝑃𝑗c_{j}^{0}=\widehat{v}_{(p)0}^{h}(P^{j}) and cj1=γj1|ωj|zv(p)j(0)superscriptsubscript𝑐𝑗1superscriptsubscript𝛾𝑗1subscript𝜔𝑗subscript𝑧subscript𝑣𝑝𝑗0c_{j}^{1}=\gamma_{j}^{-1}|\omega_{j}|\partial_{z}v_{(p)j}(0), see (3.11), we rewrite (4.36) as follows:

𝒱(p)0h=v^(p)0h+(𝒳h1)jχj(v^(p)0hv^(p)0h(Pj))(2π)1𝒳hjχjb(p)jhlnrj+jχjcj1(h𝐰^j+𝐖~j0).superscriptsubscript𝒱𝑝0superscriptsubscript^𝑣𝑝0superscript𝒳1subscript𝑗subscript𝜒𝑗superscriptsubscript^𝑣𝑝0superscriptsubscript^𝑣𝑝0superscript𝑃𝑗superscript2𝜋1superscript𝒳subscript𝑗subscript𝜒𝑗superscriptsubscript𝑏𝑝𝑗subscript𝑟𝑗subscript𝑗subscript𝜒𝑗superscriptsubscript𝑐𝑗1subscript^𝐰𝑗superscriptsubscript~𝐖𝑗0\mathcal{V}_{(p)0}^{h}=\widehat{v}_{(p)0}^{h}+(\mathcal{X}^{h}-1){\textstyle\sum\nolimits_{j}}\chi_{j}(\widehat{v}_{(p)0}^{h}-\widehat{v}_{(p)0}^{h}(P^{j}))-(2\pi)^{-1}\mathcal{X}^{h}{\textstyle\sum\nolimits_{j}}\chi_{j}b_{(p)j}^{h}\ln r_{j}+{\textstyle\sum\nolimits_{j}}\chi_{j}c_{j}^{1}(h\widehat{\mathbf{w}}_{j}+\widetilde{\mathbf{W}}_{j}^{0}).

We list the estimates

||(1𝒳h)χj(v^(p)0h\displaystyle||(1-\mathcal{X}^{h})\chi_{j}(\widehat{v}_{(p)0}^{h} v^(p)0h(Pj));H1(Ω(h))||ch3/2δ||v~(p)0h;Vδ2(ω0)||ch,\displaystyle-\widehat{v}_{(p)0}^{h}(P^{j}));H^{1}(\Omega_{\bullet}(h))||\leq ch^{3/2-\delta}||\widetilde{v}_{(p)0}^{h};V_{\delta}^{2}(\omega_{0})||\leq ch, (4.42)
||b(p)jhχjlnrj;H1(Ω(h))||\displaystyle||b_{(p)j}^{h}\chi_{j}\ln r_{j};H^{1}(\Omega_{\bullet}(h))|| ch1/2|lnh|1(hR(rj2+1)rj𝑑rj)1/2ch1/2|lnh|1/2,absent𝑐superscript12superscript1superscriptsuperscriptsubscript𝑅superscriptsubscript𝑟𝑗21subscript𝑟𝑗differential-dsubscript𝑟𝑗12𝑐superscript12superscript12\displaystyle\leq ch^{1/2}|\ln h|^{-1}\left(\int_{h}^{R}(r_{j}^{-2}+1)r_{j}dr_{j}\right)^{1/2}\leq ch^{1/2}|\ln h|^{-1/2},
h||χjcj1h𝐰^j;H1(Ω(h))||\displaystyle h||\chi_{j}c_{j}^{1}h\widehat{\mathbf{w}}_{j};H^{1}(\Omega_{\bullet}(h))|| ch3/2|lnh|1||𝐰^j;H1(Λj)||,\displaystyle\leq ch^{3/2}|\ln h|^{-1}||\widehat{\mathbf{w}}_{j};H^{1}(\Lambda_{j})||,
||cj1χj𝐖~j0;H1(Ω(h))||\displaystyle||c_{j}^{1}\chi_{j}\widetilde{\mathbf{W}}_{j}^{0};H^{1}(\Omega_{\bullet}(h))|| ch1/2|lnh|1(hR(h2rj4+1rj2)rj𝑑rj)1/2ch1/2|lnh|1/2.absent𝑐superscript12superscript1superscriptsuperscriptsubscript𝑅superscript2superscriptsubscript𝑟𝑗41superscriptsubscript𝑟𝑗2subscript𝑟𝑗differential-dsubscript𝑟𝑗12𝑐superscript12superscript12\displaystyle\leq ch^{1/2}|\ln h|^{-1}\left(\int_{h}^{R}\left(\frac{h^{2}}{r_{j}^{4}}+\frac{1}{r_{j}^{2}}\right)r_{j}dr_{j}\right)^{1/2}\leq ch^{1/2}|\ln h|^{-1/2}.

These must be commented. Notice that the factor h1/2superscript12h^{1/2} comes due to integration in z(0,h)𝑧0z\in(0,h). In the first estimate we took into account that 1𝒳h=01superscript𝒳01-\mathcal{X}^{h}=0 outside the disk 𝔹2Rh(Pj)subscript𝔹2𝑅superscript𝑃𝑗\mathbb{B}_{2Rh}(P^{j}) where rj2Rhsubscript𝑟𝑗2𝑅r_{j}\leq 2Rh and applied the weighted inequality (1.9) for v~(p)0h(y)=v^(p)0h(y)v^(p)0h(Pj)superscriptsubscript~𝑣𝑝0𝑦superscriptsubscript^𝑣𝑝0𝑦superscriptsubscript^𝑣𝑝0superscript𝑃𝑗\widetilde{v}_{(p)0}^{h}(y)=\widehat{v}_{(p)0}^{h}(y)-\widehat{v}_{(p)0}^{h}(P^{j}) with δ(0,1/2)𝛿012\delta\in(0,1/2). The second and fourth estimates were derived by a direct calculation of norms and using the decomposition (3.4) of 𝐖j0superscriptsubscript𝐖𝑗0\mathbf{W}_{j}^{0} and the bound c|lnh|1𝑐superscript1c|\ln h|^{-1} for |cj1|superscriptsubscript𝑐𝑗1|c_{j}^{1}|, cf. (4.39)-(4.41). The third estimate is obtained by the coordinate change xξjmaps-to𝑥superscript𝜉𝑗x\mapsto\xi^{j}, see (3.1).

The above listed estimates support the following relation:

|(x𝒱(p)0h,x𝒱(q)0h)Ω(h)+(𝒱(p)0h,𝒱(q)0h)Ω(h)h(yv^(p)0h,yv^(q)0h)ω(h)\displaystyle|(\nabla_{x}\mathcal{V}_{(p)0}^{h},\nabla_{x}\mathcal{V}_{(q)0}^{h})_{\Omega_{\bullet}(h)}+(\mathcal{V}_{(p)0}^{h},\mathcal{V}_{(q)0}^{h})_{\Omega_{\bullet}(h)}-h(\nabla_{y}\widehat{v}_{(p)0}^{h},\nabla_{y}\widehat{v}_{(q)0}^{h})_{\omega_{\bullet}(h)} (4.43)
h(v^(p)0h,v^(q)0h)ω(h)|c|lnh|1/2.\displaystyle-h(\widehat{v}_{(p)0}^{h},\widehat{v}_{(q)0}^{h})_{\omega_{\bullet}(h)}|\leq c|\ln h|^{-1/2}.

Moreover,

|(v^(p)0h,v^(q)0h)ω(h)(v(p)0h,v(q)0h)ω0|subscriptsuperscriptsubscript^𝑣𝑝0superscriptsubscript^𝑣𝑞0subscript𝜔subscriptsuperscriptsubscript𝑣𝑝0superscriptsubscript𝑣𝑞0subscript𝜔0\displaystyle|(\widehat{v}_{(p)0}^{h},\widehat{v}_{(q)0}^{h})_{\omega_{\bullet}(h)}-(v_{(p)0}^{h},v_{(q)0}^{h})_{\omega_{0}}| c|lnh|1/2,absent𝑐superscript12\displaystyle\leq c|\ln h|^{-1/2}, (4.44)
|(yv^(p)0h,yv^(q)0h)ω(h)μph(v(p)0h,v(q)0h)ω0|subscriptsubscript𝑦superscriptsubscript^𝑣𝑝0subscript𝑦superscriptsubscript^𝑣𝑞0subscript𝜔superscriptsubscript𝜇𝑝subscriptsuperscriptsubscript𝑣𝑝0superscriptsubscript𝑣𝑞0subscript𝜔0\displaystyle|(\nabla_{y}\widehat{v}_{(p)0}^{h},\nabla_{y}\widehat{v}_{(q)0}^{h})_{\omega_{\bullet}(h)}-\mu_{p}^{h}(v_{(p)0}^{h},v_{(q)0}^{h})_{\omega_{0}}| c|lnh|1/2.absent𝑐superscript12\displaystyle\leq c|\ln h|^{-1/2}. (4.45)

Indeed, in (4.44) we got rid of lnrjsubscript𝑟𝑗\ln r_{j} by using the second estimate (4.42) and evaluate ||v^(p)0h;L2(ωjh)||||\widehat{v}_{(p)0}^{h};L^{2}(\omega_{j}^{h})|| by means of the weighted inequality (1.9) again. To conclude (4.45), we observed additionally that

|(yv^(p)0h,yv^(q)0h)ω0μph(v^(p)0h,v^(q)0h)ω0|c|lnh|1/2subscriptsubscript𝑦superscriptsubscript^𝑣𝑝0subscript𝑦superscriptsubscript^𝑣𝑞0subscript𝜔0superscriptsubscript𝜇𝑝subscriptsuperscriptsubscript^𝑣𝑝0superscriptsubscript^𝑣𝑞0subscript𝜔0𝑐superscript12|(\nabla_{y}\widehat{v}_{(p)0}^{h},\nabla_{y}\widehat{v}_{(q)0}^{h})_{\omega_{0}}-\mu_{p}^{h}(\widehat{v}_{(p)0}^{h},\widehat{v}_{(q)0}^{h})_{\omega_{0}}|\leq c|\ln h|^{-1/2}

because v^(p)0hsuperscriptsubscript^𝑣𝑝0\widehat{v}_{(p)0}^{h} is a solution of the problem

Δv^(p)0hμphv^(p)0h=f^(p)0h in ω0,νv^(p)0h=0 on ω0formulae-sequenceΔsuperscriptsubscript^𝑣𝑝0superscriptsubscript𝜇𝑝superscriptsubscript^𝑣𝑝0superscriptsubscript^𝑓𝑝0 in subscript𝜔0subscript𝜈superscriptsubscript^𝑣𝑝00 on subscript𝜔0-\Delta\widehat{v}_{(p)0}^{h}-\mu_{p}^{h}\widehat{v}_{(p)0}^{h}=\widehat{f}_{(p)0}^{h}\text{ in }\omega_{0},\ \ \ \partial_{\nu}\widehat{v}_{(p)0}^{h}=0\text{ on }\partial\omega_{0}

with a right-hand side which is caused by abolition of b(p)jhχj(2π)1lnrjsuperscriptsubscript𝑏𝑝𝑗subscript𝜒𝑗superscript2𝜋1subscript𝑟𝑗-b_{(p)j}^{h}\chi_{j}(2\pi)^{-1}\ln r_{j} and therefore has the L2(ω0)superscript𝐿2subscript𝜔0L^{2}(\omega_{0})-norm of order |lnh|1superscript1|\ln h|^{-1}.

From (4.43)-(4.45) we derive that

|(x𝒱(p)0h,x𝒱(q)0h)Ω(h)+(𝒱(p)0h,𝒱(q)0h)Ω(h)h(1+μph)(v(p)0h,v(q)0h)ω0|cpq|lnh|1/2.subscriptsubscript𝑥superscriptsubscript𝒱𝑝0subscript𝑥superscriptsubscript𝒱𝑞0subscriptΩsubscriptsuperscriptsubscript𝒱𝑝0superscriptsubscript𝒱𝑞0subscriptΩ1superscriptsubscript𝜇𝑝subscriptsuperscriptsubscript𝑣𝑝0superscriptsubscript𝑣𝑞0subscript𝜔0subscript𝑐𝑝𝑞superscript12|(\nabla_{x}\mathcal{V}_{(p)0}^{h},\nabla_{x}\mathcal{V}_{(q)0}^{h})_{\Omega_{\bullet}(h)}+(\mathcal{V}_{(p)0}^{h},\mathcal{V}_{(q)0}^{h})_{\Omega_{\bullet}(h)}-h(1+\mu_{p}^{h})(v_{(p)0}^{h},v_{(q)0}^{h})_{\omega_{0}}|\leq c_{pq}|\ln h|^{-1/2}. (4.46)

In a similar way but with much simpler calculations (recall that vjhsuperscriptsubscript𝑣𝑗v_{j}^{h} is a smooth function on [0,lj]0subscript𝑙𝑗[0,l_{j}]) we derive the inequalities

|h1γj(x𝒱(p)jh,x𝒱(q)jh)Ωj(h)+h1ρj(𝒱(p)jh,𝒱(q)jh)Ωj(h)h(1+μph)ρ|ωj|(v(p)jh,v(q)jh)Ij|c|lnh|1/2.superscript1subscript𝛾𝑗subscriptsubscript𝑥superscriptsubscript𝒱𝑝𝑗subscript𝑥superscriptsubscript𝒱𝑞𝑗subscriptΩ𝑗superscript1subscript𝜌𝑗subscriptsuperscriptsubscript𝒱𝑝𝑗superscriptsubscript𝒱𝑞𝑗subscriptΩ𝑗1superscriptsubscript𝜇𝑝𝜌subscript𝜔𝑗subscriptsuperscriptsubscript𝑣𝑝𝑗superscriptsubscript𝑣𝑞𝑗subscript𝐼𝑗𝑐superscript12\left|h^{-1}\gamma_{j}(\nabla_{x}\mathcal{V}_{(p)j}^{h},\nabla_{x}\mathcal{V}_{(q)j}^{h})_{\Omega_{j}(h)}+h^{-1}\rho_{j}(\mathcal{V}_{(p)j}^{h},\mathcal{V}_{(q)j}^{h})_{\Omega_{j}(h)}-h(1+\mu_{p}^{h})\rho|\omega_{j}|(v_{(p)j}^{h},v_{(q)j}^{h})_{I_{j}}\right|\leq c|\ln h|^{-1/2}. (4.47)

Notice that the factor h1superscript1h^{-1} is compensated due to the relation |ωjh|=h2|ωj|superscriptsubscript𝜔𝑗superscript2subscript𝜔𝑗|\omega_{j}^{h}|=h^{2}|\omega_{j}|. According to (1.17), (4.23) and (4.32), (2.14), the inequalities (4.46) and (4.47) lead us to

|(𝒱(p)h,𝒱(q)h)hh(1+μph)δp,q|c|lnh|1/2.subscriptsuperscriptsubscript𝒱𝑝superscriptsubscript𝒱𝑞1superscriptsubscript𝜇𝑝subscript𝛿𝑝𝑞𝑐superscript12|(\mathcal{V}_{(p)}^{h},\mathcal{V}_{(q)}^{h})_{h}-h(1+\mu_{p}^{h})\delta_{p,q}|\leq c|\ln h|^{-1/2}. (4.48)

Now we evaluate the norm in (4.28), namely

δph=||Th𝒰(p)htph𝒰(p)h;Hh||=(1+μph)||𝒱(p)h;Hh||1||𝒱(p)h(1+μph)Th𝒱(p)h;Hh||.\delta_{p}^{h}=||T^{h}\mathcal{U}_{(p)}^{h}-t_{p}^{h}\mathcal{U}_{(p)}^{h};H^{h}||=(1+\mu_{p}^{h})||\mathcal{V}_{(p)}^{h};H^{h}||^{-1}||\mathcal{V}_{(p)}^{h}-(1+\mu_{p}^{h})T^{h}\mathcal{V}_{(p)}^{h};H^{h}||. (4.49)

The first factor on the right is bounded, see Theorem 14, and the second one does not exceed cph1subscript𝑐𝑝superscript1c_{p}h^{-1} owing to (4.48) with p=q𝑝𝑞p=q. Using a definition of a Hilbert norm together with formulas (4.23) and (4.24), we obtain that the last factor is equal to

sup|(𝒱(p)h(1+μph)Th𝒱(p)h,Wh)h|=sup|a(𝒱(p)h,Wh;Ξ(h))μphb(𝒱(p)h,Wh;Ξ(h))|supremumsubscriptsuperscriptsubscript𝒱𝑝1superscriptsubscript𝜇𝑝superscript𝑇superscriptsubscript𝒱𝑝superscript𝑊supremum𝑎superscriptsubscript𝒱𝑝superscript𝑊Ξsuperscriptsubscript𝜇𝑝𝑏superscriptsubscript𝒱𝑝superscript𝑊Ξ\sup\left|(\mathcal{V}_{(p)}^{h}-(1+\mu_{p}^{h})T^{h}\mathcal{V}_{(p)}^{h},W^{h})_{h}\right|=\sup\left|a(\mathcal{V}_{(p)}^{h},W^{h};\Xi(h))-\mu_{p}^{h}b(\mathcal{V}_{(p)}^{h},W^{h};\Xi(h))\right| (4.50)

where supremum is computed over the unit ball in Hhsuperscript𝐻H^{h}. Inequality (4.17), see [4, Thm. 9], indicates bounds for weighted Lebesgue norms of 𝒲hsuperscript𝒲\mathcal{W}^{h}.

We insert representations (4.36) and (4.37) into the last expressions 𝒥hsuperscript𝒥\mathcal{J}^{h} between the modulo sign in (4.50) and detach the elementary term

𝒥𝐰h=hjcj1(x𝐰^j,xWh)Ω(h)μph(𝐰^j,Wh)Ω(h),|𝒥𝐰h|ch3/2|lnh|1.formulae-sequencesuperscriptsubscript𝒥𝐰subscript𝑗superscriptsubscript𝑐𝑗1subscriptsubscript𝑥subscript^𝐰𝑗subscript𝑥superscript𝑊subscriptΩsuperscriptsubscript𝜇𝑝subscriptsubscript^𝐰𝑗superscript𝑊subscriptΩsuperscriptsubscript𝒥𝐰𝑐superscript32superscript1\mathcal{J}_{\mathbf{w}}^{h}=h{\textstyle\sum\nolimits_{j}}c_{j}^{1}(\nabla_{x}\widehat{\mathbf{w}}_{j},\nabla_{x}W^{h})_{\Omega_{\bullet}(h)}-\mu_{p}^{h}(\widehat{\mathbf{w}}_{j},W^{h})_{\Omega_{\bullet}(h)},\ \ \ |\mathcal{J}_{\mathbf{w}}^{h}|\leq ch^{3/2}|\ln h|^{-1}. (4.51)

We derived the estimate in the same way as above, that is, the information on 𝐰^j,subscript^𝐰𝑗\widehat{\mathbf{w}}_{j}, cj1superscriptsubscript𝑐𝑗1c_{j}^{1} and the coordinate change xξjmaps-to𝑥superscript𝜉𝑗x\mapsto\xi^{j}.

Other ingredients are sufficiently smooth while integration by parts and commuting the Laplace operator with cut-off functions yield

𝒥0hsuperscriptsubscript𝒥0\displaystyle\mathcal{J}_{0}^{h} =(Δxv(p)0h+μphv(p)0h,𝒳hWh)Ω(h)+([Δx,𝒳h]v~(p)0h,W0h)Ω(h),absentsubscriptsubscriptΔ𝑥superscriptsubscript𝑣𝑝0superscriptsubscript𝜇𝑝superscriptsubscript𝑣𝑝0superscript𝒳superscript𝑊subscriptΩsubscriptsubscriptΔ𝑥superscript𝒳superscriptsubscript~𝑣𝑝0superscriptsubscript𝑊0subscriptΩ\displaystyle=(\Delta_{x}v_{(p)0}^{h}+\mu_{p}^{h}v_{(p)0}^{h},\mathcal{X}^{h}W^{h})_{\Omega_{\bullet}(h)}+([\Delta_{x},\mathcal{X}^{h}]\widetilde{v}_{(p)0}^{h},W_{0}^{h})_{\Omega_{\bullet}(h)}, (4.52)
𝒥j0hsuperscriptsubscript𝒥𝑗0\displaystyle\mathcal{J}_{j0}^{h} =(Δx𝐖j0,χjW0h)Ω(h)+([Δx,χj]𝐖~j0,W0h)Ω(h)absentsubscriptsubscriptΔ𝑥superscriptsubscript𝐖𝑗0subscript𝜒𝑗superscriptsubscript𝑊0subscriptΩsubscriptsubscriptΔ𝑥subscript𝜒𝑗superscriptsubscript~𝐖𝑗0superscriptsubscript𝑊0subscriptΩ\displaystyle=(\Delta_{x}\mathbf{W}_{j}^{0},\chi_{j}W_{0}^{h})_{\Omega_{\bullet}(h)}+([\Delta_{x},\chi_{j}]\widetilde{\mathbf{W}}_{j}^{0},W_{0}^{h})_{\Omega_{\bullet}(h)} (4.53)
+cj1μph(𝐖j0+𝒳h(2π)1(ln|η|lnclog(ωj)),χjWh)Ω(h),superscriptsubscript𝑐𝑗1superscriptsubscript𝜇𝑝subscriptsuperscriptsubscript𝐖𝑗0superscript𝒳superscript2𝜋1𝜂subscript𝑐subscript𝜔𝑗subscript𝜒𝑗superscript𝑊subscriptΩ\displaystyle+c_{j}^{1}\mu_{p}^{h}(\mathbf{W}_{j}^{0}+\mathcal{X}^{h}(2\pi)^{-1}(\ln|\eta|-\ln c_{\log}(\omega_{j})),\chi_{j}W^{h})_{\Omega_{\bullet}(h)},
𝒥jhsuperscriptsubscript𝒥𝑗\displaystyle\mathcal{J}_{j}^{h} =h1(γjΔxv(p)jh+μphρjv(p)jh,XhWh)Ωj(h)+h1γj([Δx,Xh]v~(p)jh,Wh)Ωj(h),absentsuperscript1subscriptsubscript𝛾𝑗subscriptΔ𝑥superscriptsubscript𝑣𝑝𝑗superscriptsubscript𝜇𝑝subscript𝜌𝑗superscriptsubscript𝑣𝑝𝑗superscript𝑋superscript𝑊subscriptΩ𝑗superscript1subscript𝛾𝑗subscriptsubscriptΔ𝑥superscript𝑋superscriptsubscript~𝑣𝑝𝑗superscript𝑊subscriptΩ𝑗\displaystyle=h^{-1}(\gamma_{j}\Delta_{x}v_{(p)j}^{h}+\mu_{p}^{h}\rho_{j}v_{(p)j}^{h},X^{h}W^{h})_{\Omega_{j}(h)}+h^{-1}\gamma_{j}([\Delta_{x},X^{h}]\widetilde{v}_{(p)j}^{h},W^{h})_{\Omega_{j}(h)}, (4.54)
𝒥jjhsuperscriptsubscript𝒥𝑗𝑗\displaystyle\mathcal{J}_{jj}^{h} =cj1γj(Δx𝐖j,XjWh)+cj1γj([Δx,Xj]𝐖~j,Wh)Ωj(h)absentsuperscriptsubscript𝑐𝑗1subscript𝛾𝑗subscriptΔ𝑥subscript𝐖𝑗subscript𝑋𝑗superscript𝑊superscriptsubscript𝑐𝑗1subscript𝛾𝑗subscriptsubscriptΔ𝑥subscript𝑋𝑗subscript~𝐖𝑗superscript𝑊subscriptΩ𝑗\displaystyle=c_{j}^{1}\gamma_{j}(\Delta_{x}\mathbf{W}_{j},X_{j}W^{h})+c_{j}^{1}\gamma_{j}([\Delta_{x},X_{j}]\widetilde{\mathbf{W}}_{j},W^{h})_{\Omega_{j}(h)} (4.55)
+cj1μphρj(𝐖jXhγj1|ωj|1ζj,Wh)Ωj(h).superscriptsubscript𝑐𝑗1superscriptsubscript𝜇𝑝subscript𝜌𝑗subscriptsubscript𝐖𝑗superscript𝑋superscriptsubscript𝛾𝑗1superscriptsubscript𝜔𝑗1superscript𝜁𝑗superscript𝑊subscriptΩ𝑗\displaystyle+c_{j}^{1}\mu_{p}^{h}\rho_{j}(\mathbf{W}_{j}-X^{h}\gamma_{j}^{-1}|\omega_{j}|^{-1}\zeta^{j},W^{h})_{\Omega_{j}(h)}.

Note that integrals over the surfaces υ0(h),subscript𝜐0\upsilon_{0}(h), ωj0(0)superscriptsubscript𝜔𝑗00\omega_{j}^{0}(0) and υj(h)subscript𝜐𝑗\upsilon_{j}(h) vanish due to our choice of cut-off functions and boundary conditions for v0h,superscriptsubscript𝑣0v_{0}^{h}, vjhsuperscriptsubscript𝑣𝑗v_{j}^{h} and 𝐖j0,superscriptsubscript𝐖𝑗0\mathbf{W}_{j}^{0}, 𝐖jsubscript𝐖𝑗\mathbf{W}_{j}, see Section 1.2 and 3.1, respectively. We will estimate all scalar products in (4.52)-(4.54) and explain how their sum converts into 𝒥h𝒥𝐰hsuperscript𝒥superscriptsubscript𝒥𝐰\mathcal{J}^{h}-\mathcal{J}_{\mathbf{w}}^{h}.

In view of the equation (1.20) the first term in (4.52) is null. The next term 𝒥0h2superscriptsubscript𝒥02\mathcal{J}_{0}^{h2} in (4.52) is obtained from the first and third terms in (4.36) after commuting the Laplace operator with the cut-off function 𝒳hsuperscript𝒳\mathcal{X}^{h}; notice that

[Δx,𝒳hχj]subscriptΔ𝑥superscript𝒳subscript𝜒𝑗\displaystyle[\Delta_{x},\mathcal{X}^{h}\chi_{j}] =[Δx,𝒳h]+[Δx,χj],absentsubscriptΔ𝑥superscript𝒳subscriptΔ𝑥subscript𝜒𝑗\displaystyle=[\Delta_{x},\mathcal{X}^{h}]+[\Delta_{x},\chi_{j}], (4.56)
[Δx,𝒳h]subscriptΔ𝑥superscript𝒳\displaystyle[\Delta_{x},\mathcal{X}^{h}] =2x𝒳hx+Δx𝒳h,[Δx,χj]=2xχjx+Δxχj.formulae-sequenceabsent2subscript𝑥superscript𝒳subscript𝑥subscriptΔ𝑥superscript𝒳subscriptΔ𝑥subscript𝜒𝑗2subscript𝑥subscript𝜒𝑗subscript𝑥subscriptΔ𝑥subscript𝜒𝑗\displaystyle=2\nabla_{x}\mathcal{X}^{h}\cdot\nabla_{x}+\Delta_{x}\mathcal{X}^{h},\ \ \ [\Delta_{x},\chi_{j}]=2\nabla_{x}\chi_{j}\cdot\nabla_{x}+\Delta_{x}\chi_{j}.

Since v~(p)0h(Pj)=0superscriptsubscript~𝑣𝑝0superscript𝑃𝑗0\widetilde{v}_{(p)0}^{h}(P^{j})=0, see (2.9), and supports of coefficients in the differential operator [Δx,𝒳h]subscriptΔ𝑥superscript𝒳[\Delta_{x},\mathcal{X}^{h}] belong to the disk 𝔹2Rh(Pj)subscript𝔹2𝑅superscript𝑃𝑗\mathbb{B}_{2Rh}(P^{j}), see (4.34), the direct consequence of the one-dimensional Hardy inequality

||rj2(1+|lnrj|)1v~(p)0h;L2(𝔹2Rh(Pj))||\displaystyle||r_{j}^{-2}(1+|\ln r_{j}|)^{-1}\widetilde{v}_{(p)0}^{h};L^{2}(\mathbb{B}_{2Rh}(P^{j}))|| c||rj2(1+|lnrj|)1yv~(p)0h;L2(𝔹2Rh(Pj))||\displaystyle\leq c||r_{j}^{-2}(1+|\ln r_{j}|)^{-1}\nabla_{y}\widetilde{v}_{(p)0}^{h};L^{2}(\mathbb{B}_{2Rh}(P^{j}))|| (4.57)
||v~(p)0h;H2(𝔹2Rh(Pj))||\displaystyle\leq||\widetilde{v}_{(p)0}^{h};H^{2}(\mathbb{B}_{2Rh}(P^{j}))||

provides that

|𝒥0h1|(h2h2(1+|lnh|)+h1h(1+|lnh|))h1/2||v~(p)0h;H2(𝔹2Rh(Pj))||\displaystyle|\mathcal{J}_{0}^{h1}|\leq(h^{-2}h^{2}(1+|\ln h|)+h^{-1}h(1+|\ln h|))h^{1/2}||\widetilde{v}_{(p)0}^{h};H^{2}(\mathbb{B}_{2Rh}(P^{j}))|| (4.58)
×h(1+|lnh|)||r1(1+|lnr|)1W0h;L2(Ω(h))||ch3/2|lnh|3.\displaystyle\times h(1+|\ln h|)||r^{-1}(1+|\ln r|)^{-1}W_{0}^{h};L^{2}(\Omega_{\bullet}(h))||\leq ch^{3/2}|\ln h|^{3}.

Here, we took into account that r=rj<2Rh𝑟subscript𝑟𝑗2𝑅r=r_{j}<2Rh for y𝔹2Rh(Pj)𝑦subscript𝔹2𝑅superscript𝑃𝑗y\in\mathbb{B}_{2Rh}(P^{j}). The first (long) multiplier in the middle of (4.58) is written according to the relation |yk𝒳h(y)|Ckhksuperscriptsubscript𝑦𝑘superscript𝒳𝑦subscript𝐶𝑘superscript𝑘|\nabla_{y}^{k}\mathcal{X}^{h}(y)|\leq C_{k}h^{-k}, formula for [Δx,𝒳h]subscriptΔ𝑥superscript𝒳[\Delta_{x},\mathcal{X}^{h}] in (4.56) and weights in (4.57). The factor h1/2superscript12h^{1/2} is due to integration in z(0,h)𝑧0z\in(0,h) and finally the weights r1(1+|lnr|)1superscript𝑟1superscript1𝑟1r^{-1}(1+|\ln r|)^{-1} and (1+|lnh|)1superscript11(1+|\ln h|)^{-1} from (4.17) were considered. It should be also mentioned that h0<1subscript01h_{0}<1 and, therefore, |lnh|c>0𝑐0|\ln h|\geq c>0 for h(0,h0]0subscript0h\in(0,h_{0}].

Dealing with (4.53) we observe that the Laplace equation for 𝐖j0superscriptsubscript𝐖𝑗0\mathbf{W}_{j}^{0} in (3.3) annuls the first term 𝒥j0h1superscriptsubscript𝒥𝑗01\mathcal{J}_{j0}^{h1}. Supports of coefficients of [Δx,χj]subscriptΔ𝑥subscript𝜒𝑗[\Delta_{x},\mathcal{\chi}_{j}], see (4.56), are located in the annulus {xΩ0(h)¯:RjrjRj+}conditional-set𝑥¯subscriptΩ0superscriptsubscript𝑅𝑗subscript𝑟𝑗superscriptsubscript𝑅𝑗\{x\in\overline{\Omega_{0}(h)}:R_{j}^{-}\leq r_{j}\leq R_{j}^{+}\}. Hence, using the decay rate in |ηj|=ρj=h1rjsuperscript𝜂𝑗superscript𝜌𝑗superscript1subscript𝑟𝑗|\eta^{j}|=\rho^{j}=h^{-1}r_{j} of the remainder 𝐖~j0superscriptsubscript~𝐖𝑗0\widetilde{\mathbf{W}}_{j}^{0}, see (3.4), we derive the following estimates for the second and third terms in (4.53):

|𝒥0jh2|superscriptsubscript𝒥0𝑗2\displaystyle|\mathcal{J}_{0j}^{h2}| c|cj1|h1/2(RjRj+(1h21(1+ρj)4+1(1+ρj)2)rjdrj)1/2||Wh;L2(Ω(h))||ch3/2,\displaystyle\leq c|c_{j}^{1}|h^{1/2}\left(\int_{R_{j}^{-}}^{R_{j}^{+}}\left(\frac{1}{h^{2}}\frac{1}{(1+\rho_{j})^{4}}+\frac{1}{(1+\rho_{j})^{2}}\right)r_{j}dr_{j}\right)^{1/2}||W^{h};L^{2}(\Omega_{\bullet}(h))||\leq ch^{3/2}, (4.59)
|𝒥0jh3|superscriptsubscript𝒥0𝑗3\displaystyle|\mathcal{J}_{0j}^{h3}| c|cj1|h1/2(RhRj+rjdrj(1+ρj)2)1/2||Wh;L2(Ω(h))||ch3/2|lnh|.\displaystyle\leq c|c_{j}^{1}|h^{1/2}\left(\int_{Rh}^{R_{j}^{+}}\frac{r_{j}dr_{j}}{(1+\rho_{j})^{2}}\right)^{1/2}||W^{h};L^{2}(\Omega_{\bullet}(h))||\leq ch^{3/2}|\ln h|.

Here, we applied (4.17) again and recalled that |cj1|c|lnh|1superscriptsubscript𝑐𝑗1𝑐superscript1|c_{j}^{1}|\leq c|\ln h|^{-1}. It should be mentioned that 𝒥0jh2superscriptsubscript𝒥0𝑗2\mathcal{J}_{0j}^{h2} and 𝒥0jh3superscriptsubscript𝒥0𝑗3\mathcal{J}_{0j}^{h3}, respectively, involve the commutator [Δx,χj]subscriptΔ𝑥subscript𝜒𝑗[\Delta_{x},\mathcal{\chi}_{j}] and the multiplication operator μphsuperscriptsubscript𝜇𝑝\mu_{p}^{h} acting on cj1𝐖j0superscriptsubscript𝑐𝑗1superscriptsubscript𝐖𝑗0c_{j}^{1}\mathbf{W}_{j}^{0} and the last term in (4.36). In this way the sum 𝒥0h+𝒥01h++𝒥0Jh+𝒥𝐰hsuperscriptsubscript𝒥0superscriptsubscript𝒥01superscriptsubscript𝒥0𝐽superscriptsubscript𝒥𝐰\mathcal{J}_{0}^{h}+\mathcal{J}_{01}^{h}+...+\mathcal{J}_{0J}^{h}+\mathcal{J}_{\mathbf{w}}^{h} exhibits the whole part of 𝒥hsuperscript𝒥\mathcal{J}^{h} generated by (4.36).

Referring to (4.54), we see that 𝒥jh1=0superscriptsubscript𝒥𝑗10\mathcal{J}_{j}^{h1}=0 in view of (1.21). The second term 𝒥jh2superscriptsubscript𝒥𝑗2\mathcal{J}_{j}^{h2} in (4.54) meets the estimate

|𝒥jh2|ch1(h2h2+h1h)h3/2||Wjh;L2(Ωjh)||ch3/2.|\mathcal{J}_{j}^{h2}|\leq ch^{-1}(h^{-2}h^{2}+h^{-1}h)h^{3/2}||W_{j}^{h};L^{2}(\Omega_{j}^{h})||\leq ch^{3/2}.

Here, h1superscript1h^{-1} came from (4.54), the relations |xkXh(z)|ckhksuperscriptsubscript𝑥𝑘superscript𝑋𝑧subscript𝑐𝑘superscript𝑘|\nabla_{x}^{k}X^{h}(z)|\leq c_{k}h^{-k} and v~(p)jh(z)=v(p)jh(z)v(p)jh(0)zzv(p)jh(0)=O(z2)superscriptsubscript~𝑣𝑝𝑗𝑧superscriptsubscript𝑣𝑝𝑗𝑧superscriptsubscript𝑣𝑝𝑗0𝑧subscript𝑧superscriptsubscript𝑣𝑝𝑗0𝑂superscript𝑧2\widetilde{v}_{(p)j}^{h}(z)=v_{(p)j}^{h}(z)-v_{(p)j}^{h}(0)-z\partial_{z}v_{(p)j}^{h}(0)=O(z^{2}) were used, the factor h3/2superscript32h^{3/2} is due to integration over Ωjh={xΩj(h)¯:z3h}superscriptsubscriptΩ𝑗conditional-set𝑥¯subscriptΩ𝑗𝑧3superset-ofabsent\Omega_{j}^{h}=\{x\in\overline{\Omega_{j}(h)}:z\leq 3h\}\supsetsuppzXhsubscript𝑧superscript𝑋\partial_{z}X^{h} and finally the direct consequence of the Newton-Leibnitz formula

h2||Wjh;L2(Ωjh)||2ch1(||zWjh;L2(Ωjh)||2+||Wjh;L2(Ωjh)||2)c||Wjh;Hh||2=ch^{-2}||W_{j}^{h};L^{2}(\Omega_{j}^{h})||^{2}\leq ch^{-1}(||\partial_{z}W_{j}^{h};L^{2}(\Omega_{j}^{h})||^{2}+||W_{j}^{h};L^{2}(\Omega_{j}^{h})||^{2})\leq c||W_{j}^{h};H^{h}||^{2}=c (4.60)

together with the inequality (4.17) were applied.

Similarly to the above considerations, the first term 𝒥jjh1superscriptsubscript𝒥𝑗𝑗1\mathcal{J}_{jj}^{h1} in (4.55) vanishes, cf. (3.2), and the other couple of terms can be estimated as follows:

|𝒥jjh2|+|𝒥jjh3|superscriptsubscript𝒥𝑗𝑗2superscriptsubscript𝒥𝑗𝑗3\displaystyle|\mathcal{J}_{jj}^{h2}|+|\mathcal{J}_{jj}^{h3}| c|cj1|(Ωjhdx||Wjh;L2(Ωjh)||2+Ωj(h)e2δz/hdx||Wjh;L2(Ωj(h))||2)1/2\displaystyle\leq c|c_{j}^{1}|\left(\int_{\Omega_{j}^{h}}dx||W_{j}^{h};L^{2}(\Omega_{j}^{h})||^{2}+\int_{\Omega_{j}(h)}e^{-2\delta z/h}dx||W_{j}^{h};L^{2}(\Omega_{j}(h))||^{2}\right)^{1/2}
c|lnh|1(h3h2+h3h)1/2c|lnh|1h2.absent𝑐superscript1superscriptsuperscript3superscript2superscript312𝑐superscript1superscript2\displaystyle\leq c|\ln h|^{-1}(h^{3}h^{2}+h^{3}h)^{1/2}\leq c|\ln h|^{-1}h^{2}.

Here, we took into account the exponential decay in (3.6) together with formulas (4.60) and (4.17).

In the same way as above we detect a proper redistribution of commutators of ΔxsubscriptΔ𝑥\Delta_{x} with Xh,superscript𝑋X^{h}, Xjsubscript𝑋𝑗X_{j} and conclude that 𝒥jh+𝒥jjhsuperscriptsubscript𝒥𝑗superscriptsubscript𝒥𝑗𝑗\mathcal{J}_{j}^{h}+\mathcal{J}_{jj}^{h} equals a part of 𝒥hsuperscript𝒥\mathcal{J}^{h} generated by (4.37).

We summarize our calculations and find that the worst bound in estimates derived for components of (4.50), occurs in (4.58). Hence, according to (4.48) with q=p𝑞𝑝q=p, we obtain the following inequality for the quantity (4.49):

δphcph1/2|lnh|3.superscriptsubscript𝛿𝑝subscript𝑐𝑝superscript12superscript3\delta_{p}^{h}\leq c_{p}h^{1/2}|\ln h|^{3}.

Lemma 15 gives us eigenvalues τn(h)superscript𝜏𝑛\tau^{n}(h) and λn(h)=τn(h)11superscript𝜆𝑛superscript𝜏𝑛superscript11\lambda^{n}(h)=\tau^{n}(h)^{-1}-1, see (4.26), of the problems (4.25) and (1.6)-(1.12), respectively, such that, by virtue of (4.33),

τn(h)superscript𝜏𝑛\displaystyle\tau^{n}(h) [tphcph1/2|lnh|3,tph+cph1/2|lnh|3]absentsuperscriptsubscript𝑡𝑝subscript𝑐𝑝superscript12superscript3superscriptsubscript𝑡𝑝subscript𝑐𝑝superscript12superscript3\displaystyle\in[t_{p}^{h}-c_{p}h^{1/2}|\ln h|^{3},t_{p}^{h}+c_{p}h^{1/2}|\ln h|^{3}] (4.61)
λn(h)[μphCph1/2|lnh|3,μph+Cph1/2|lnh|3]h(0,hp)formulae-sequenceabsentsuperscript𝜆𝑛superscriptsubscript𝜇𝑝subscript𝐶𝑝superscript12superscript3superscriptsubscript𝜇𝑝subscript𝐶𝑝superscript12superscript3for-all0subscript𝑝\displaystyle\Rightarrow\lambda^{n}(h)\in\left[\mu_{p}^{h}-C_{p}h^{1/2}|\ln h|^{3},\mu_{p}^{h}+C_{p}h^{1/2}|\ln h|^{3}\right]\ \ \ \ \forall h\in(0,h_{p})

where cp,subscript𝑐𝑝c_{p}, Cpsubscript𝐶𝑝C_{p} and hpsubscript𝑝h_{p} are some positive numbers and the index n𝑛n may depend on hh.

4.5 Theorem on asymptotics

We are in position to conclude with the main assertion of the paper.

Theorem 16

For any N𝑁N\in\mathbb{N}, there exist positive h(N)𝑁h(N) and c(N)𝑐𝑁c(N) such that the entries λ1(h),,λN(h)superscript𝜆1superscript𝜆𝑁\lambda^{1}(h),...,\lambda^{N}(h) of the eigenvalue sequence (1.18) of the original problem (1.6)-(1.14) (or (1.16) in the variational formulation) and the first N𝑁N positive eigenvalues

0<μ1hμ2hμNh0superscriptsubscript𝜇1superscriptsubscript𝜇2superscriptsubscript𝜇𝑁0<\mu_{1}^{h}\leq\mu_{2}^{h}\leq...\leq\mu_{N}^{h} (4.62)

of the equation (3.14) or (3.15) are in the relationship

|λn(h)μnh|c(N)h1/2|lnh|3,n=1,,N,h(0,h(N)).formulae-sequencesuperscript𝜆𝑛superscriptsubscript𝜇𝑛𝑐𝑁superscript12superscript3formulae-sequence𝑛1𝑁0𝑁|\lambda^{n}(h)-\mu_{n}^{h}|\leq c(N)h^{1/2}|\ln h|^{3},\ \ n=1,...,N,\ h\in(0,h(N)). (4.63)

Proof. The result directly stems from (4.61) and all the previous considerations. We still need to utter several important remarks, in order to complete the proof. First, if μphsuperscriptsubscript𝜇𝑝\mu_{p}^{h} is eigenvalue of multiplicity ϰph1superscriptsubscriptitalic-ϰ𝑝1\varkappa_{p}^{h}\geq 1, then Lemma 15 provides us with ϰphsuperscriptsubscriptitalic-ϰ𝑝\varkappa_{p}^{h} different eigenvalues λn(h),,λn+ϰph1(h)superscript𝜆𝑛superscript𝜆𝑛superscriptsubscriptitalic-ϰ𝑝1\lambda^{n}(h),...,\lambda^{n+\varkappa_{p}^{h}-1}(h) satisfying (4.61) with a bigger constant Cpsubscript𝐶𝑝C_{p}. Indeed, owing to (4.48), the constructed approximate eigenfunctions 𝒰(p)h,,𝒰(p+ϰph1)hsuperscriptsubscript𝒰𝑝superscriptsubscript𝒰𝑝superscriptsubscriptitalic-ϰ𝑝1\mathcal{U}_{(p)}^{h},...,\mathcal{U}_{(p+\varkappa_{p}^{h}-1)}^{h} are almost orthonormalized, that is

|(𝒰(k)h,𝒰(q)h)hδp,q|cp|lnh|1/2,k,q=p,,p+ϰph1.formulae-sequencesubscriptsuperscriptsubscript𝒰𝑘superscriptsubscript𝒰𝑞subscript𝛿𝑝𝑞subscript𝑐𝑝superscript12𝑘𝑞𝑝𝑝superscriptsubscriptitalic-ϰ𝑝1|(\mathcal{U}_{(k)}^{h},\mathcal{U}_{(q)}^{h})_{h}-\delta_{p,q}|\leq c_{p}|\ln h|^{-1/2},\ \ k,q=p,...,p+\varkappa_{p}^{h}-1.

Moreover, setting δh=Rmax{δph,,δp+ϰph1h}superscriptsubscript𝛿𝑅superscriptsubscript𝛿𝑝superscriptsubscript𝛿𝑝superscriptsubscriptitalic-ϰ𝑝1\delta_{\ast}^{h}=R\max\{\delta_{p}^{h},...,\delta_{p+\varkappa_{p}^{h}-1}^{h}\} in (4.30), we see that their projection onto the linear hull (uNhh,,uNh+Xh1h)superscriptsubscript𝑢superscript𝑁superscriptsubscript𝑢superscript𝑁superscript𝑋1\mathcal{L}(u_{N^{h}}^{h},...,u_{N^{h}+X^{h}-1}^{h}) are linear independent for a small hh and a big R𝑅R that is possible in the case Xhϰphsuperscript𝑋superscriptsubscriptitalic-ϰ𝑝X^{h}\geq\varkappa_{p}^{h} only. Thus, changing CpRCpmaps-tosubscript𝐶𝑝𝑅subscript𝐶𝑝C_{p}\mapsto RC_{p} in (4.61) gives us at least ϰphsuperscriptsubscriptitalic-ϰ𝑝\varkappa_{p}^{h} desired eigenvalues.

Second, since each eigenvalue μphsuperscriptsubscript𝜇𝑝\mu_{p}^{h} in (4.62) has an eigenvalue λnh(p)(h)superscript𝜆superscript𝑛𝑝\lambda^{n^{h}(p)}(h) in its small neighborhood and n(p)n(q)𝑛𝑝𝑛𝑞n(p)\neq n(q) for pq𝑝𝑞p\neq q, we obtain λn(h)μnh+cnh1/2|lnh|3superscript𝜆𝑛superscriptsubscript𝜇𝑛subscript𝑐𝑛superscript12superscript3\lambda^{n}(h)\leq\mu_{n}^{h}+c_{n}h^{1/2}|\ln h|^{3} and confirm the assumption (4.10) so that Theorem 14 becomes true.

Finally, we assume without loss of generality that N𝑁N is fixed such that the eigenvalues μN0superscriptsubscript𝜇𝑁0\mu_{N}^{0} and μN+10superscriptsubscript𝜇𝑁10\mu_{N+1}^{0} of the operator 𝒜0superscript𝒜0\mathcal{A}^{0} obey the relation μN0<μN+10superscriptsubscript𝜇𝑁0superscriptsubscript𝜇𝑁10\mu_{N}^{0}<\mu_{N+1}^{0}. If it happens that the index nh(N)superscript𝑛𝑁n^{h}(N) of the eigenvalue λnh(N)superscript𝜆superscript𝑛𝑁\lambda^{n^{h}(N)} in the vicinity of μNhsuperscriptsubscript𝜇𝑁\mu_{N}^{h} is strictly bigger that N𝑁N, then, for an infinitesimal positive sequence {hk}ksubscriptsubscript𝑘𝑘\{h_{k}\}_{k\in\mathbb{N}}, we have λN+1(hk)μN+10εsuperscript𝜆𝑁1subscript𝑘superscriptsubscript𝜇𝑁10𝜀\lambda^{N+1}(h_{k})\leq\mu_{N+1}^{0}-\varepsilon with some ε>0𝜀0\varepsilon>0. We can apply Theorem 14 and conclude that λN+1(0)=limλN+1(hk)superscript𝜆𝑁10superscript𝜆𝑁1subscript𝑘\lambda^{N+1}(0)=\lim\lambda^{N+1}(h_{k}) as k+0𝑘0k\rightarrow+0 is an eigenvalue in the interval (0,μN+10ε]0superscriptsubscript𝜇𝑁10𝜀(0,\mu_{N+1}^{0}-\varepsilon], while the limits of (4.11), (4.12) constructed from the corresponding eigenfunctions uN+1(hk,)superscript𝑢𝑁1subscript𝑘u^{N+1}(h_{k},\cdot) are orthogonal in \mathcal{L} to the vector eigenfunctions (v0p,v1p,,vJp)superscriptsubscript𝑣0𝑝superscriptsubscript𝑣1𝑝superscriptsubscript𝑣𝐽𝑝(v_{0}^{p},v_{1}^{p},...,v_{J}^{p}), p=1,,N,𝑝1𝑁p=1,...,N, of the operator 𝒜0superscript𝒜0\mathcal{A}^{0}. Since λN+1(0)superscript𝜆𝑁10\lambda^{N+1}(0) belongs to the spectrum of 𝒜0superscript𝒜0\mathcal{A}^{0} but λN+1(0)μN0superscript𝜆𝑁10superscriptsubscript𝜇𝑁0\lambda^{N+1}(0)\leq\mu_{N}^{0}, the latter is absurd. \blacksquare

Remark 17

A proof of Proposition 12 may follow the same scheme and meets crucial simplifications because, first, the total multiplicity of the negative spectrum is known a priori and, second, the corresponding approximate eigenfunctions (3.18), (3.19) decay exponentially at a distance from the points P1,,PJsuperscript𝑃1superscript𝑃𝐽P^{1},...,P^{J}.

5 Final remarks

5.1 Simplified and rough asymptotics

Since the point condition (2.28) contains the big parameter lnh\ln h in the matrix (3.13), it is straightforward to write an asymptotics in |lnh||\ln h| for eigenvalues (1.18). In view of the precision estimate (4.63) in Theorem 16 it suffices to find a decomposition of eigenvalues in the model problem (1.20)-(1.23), (2.28), (2.29), in particular, of the projections (2.15) of the corresponding eigenvectors.

Let us demonstrate the simplest ansatz for the first eigenvalue

λ1(h)μ1(h)|lnh|1μ11+|lnh|2μ21+similar-tosuperscript𝜆1superscript𝜇1similar-tosuperscript1superscriptsubscript𝜇11superscript2superscriptsubscript𝜇21\lambda^{1}(h)\sim\mu^{1}(h)\sim|\ln h|^{-1}\mu_{1}^{1}+|\ln h|^{-2}\mu_{2}^{1}+... (5.1)

The corresponding eigenvector of the model problem is searched in the asymptotic form

v1(h,)v(0)1+|lnh|1v(1)1+|lnh|2v(2)1+similar-tosuperscript𝑣1superscriptsubscript𝑣01superscript1superscriptsubscript𝑣11superscript2superscriptsubscript𝑣21v^{1}(h,\cdot)\sim v_{(0)}^{1}+|\ln h|^{-1}v_{(1)}^{1}+|\ln h|^{-2}v_{(2)}^{1}+... (5.2)

while the absence in (5.1) of the term |lnh|0μ01superscript0superscriptsubscript𝜇01|\ln h|^{0}\mu_{0}^{1} clearly requires that

v(0)1=(1,0,,0)superscriptsubscript𝑣01100v_{(0)}^{1}=(1,0,...,0) (5.3)

and therefore

+v(0)1=ε=(1,,1)J,+′′v(0)1=v(0)1=′′v(0)1=0J.formulae-sequencesuperscriptsubscriptWeierstrass-psuperscriptsubscript𝑣01𝜀11superscript𝐽superscriptsubscriptWeierstrass-p′′superscriptsubscript𝑣01superscriptsubscriptWeierstrass-psuperscriptsubscript𝑣01superscriptsubscriptWeierstrass-p′′superscriptsubscript𝑣010superscript𝐽\wp_{+}^{\prime}v_{(0)}^{1}=\varepsilon=(1,...,1)\in\mathbb{R}^{J},\ \ \wp_{+}^{\prime\prime}v_{(0)}^{1}=\wp_{-}^{\prime}v_{(0)}^{1}=\wp_{-}^{\prime\prime}v_{(0)}^{1}=0\in\mathbb{R}^{J}. (5.4)

Hence, a problem to determine v(1)1=(v(1)01,v(1)11,,v(1)J1)superscriptsubscript𝑣11superscriptsubscript𝑣101superscriptsubscript𝑣111superscriptsubscript𝑣1𝐽1v_{(1)}^{1}=(v_{(1)0}^{1},v_{(1)1}^{1},...,v_{(1)J}^{1}) involves the Neumann problem

Δyv(1)01(y)=μ11v(1)01(y)=μ11,yω,νv(1)01(y)=0,yω0,formulae-sequencesubscriptΔ𝑦superscriptsubscript𝑣101𝑦superscriptsubscript𝜇11superscriptsubscript𝑣101𝑦superscriptsubscript𝜇11formulae-sequence𝑦subscript𝜔direct-productformulae-sequencesubscript𝜈superscriptsubscript𝑣101𝑦0𝑦subscript𝜔0-\Delta_{y}v_{(1)0}^{1}(y)=\mu_{1}^{1}v_{(1)0}^{1}(y)=\mu_{1}^{1},\ \ y\in\omega_{\odot},\ \ \ \ \partial_{\nu}v_{(1)0}^{1}(y)=0,\ \ y\in\partial\omega_{0}, (5.5)

which is derived by inserting (5.1)-(5.3) into the equations (1.20)-(1.22) and extracting terms of order |lnh|1superscript1|\ln h|^{-1}. Furthermore, we obtain from the point condition (2.28) with the matrix Sh=(2π)1|lnh|𝕀+superscript𝑆superscript2𝜋1𝕀S^{h}=(2\pi)^{-1}|\ln h|\mathbb{I}+... from (3.13) that

+v(0)1+(2π)1v(1)1=0v(1)1=2πε.superscriptsubscriptWeierstrass-psuperscriptsubscript𝑣01superscript2𝜋1superscriptsubscriptWeierstrass-psuperscriptsubscript𝑣110superscriptsubscriptWeierstrass-psuperscriptsubscript𝑣112𝜋𝜀\wp_{+}^{\prime}v_{(0)}^{1}+(2\pi)^{-1}\wp_{-}^{\prime}v_{(1)}^{1}=0\Rightarrow\wp_{-}^{\prime}v_{(1)}^{1}=-2\pi\varepsilon. (5.6)

The generalized Green formula (2.16) delivers the compatibility condition in this problem, namely

μ11|ω0|superscriptsubscript𝜇11subscript𝜔0\displaystyle\mu_{1}^{1}|\omega_{0}| =(Δyv(1)01,v(0)01)ω0+(v(1)01,Δyv(0)01)ω0=+v(1)1,v(0)1v(1)1,+v(0)1absentsubscriptsubscriptΔ𝑦superscriptsubscript𝑣101superscriptsubscript𝑣001subscript𝜔0subscriptsuperscriptsubscript𝑣101subscriptΔ𝑦superscriptsubscript𝑣001subscript𝜔0subscriptWeierstrass-psuperscriptsubscript𝑣11subscriptWeierstrass-psuperscriptsubscript𝑣01subscriptWeierstrass-psuperscriptsubscript𝑣11subscriptWeierstrass-psuperscriptsubscript𝑣01\displaystyle=-(\Delta_{y}v_{(1)0}^{1},v_{(0)0}^{1})_{\omega_{0}}+(v_{(1)0}^{1},\Delta_{y}v_{(0)0}^{1})_{\omega_{0}}=\left\langle\wp_{+}v_{(1)}^{1},\wp_{-}v_{(0)}^{1}\right\rangle-\left\langle\wp_{-}v_{(1)}^{1},\wp_{+}v_{(0)}^{1}\right\rangle
=v(1)1,+v(0)1=2πε,ε=2πJ.absentsuperscriptsubscriptWeierstrass-psuperscriptsubscript𝑣11superscriptsubscriptWeierstrass-psuperscriptsubscript𝑣012𝜋𝜀𝜀2𝜋𝐽\displaystyle=-\left\langle\wp_{-}^{\prime}v_{(1)}^{1},\wp_{+}^{\prime}v_{(0)}^{1}\right\rangle=2\pi\left\langle\varepsilon,\varepsilon\right\rangle=2\pi J.

Note that we obtained in the ansatz (5.1) the first term

μ11=2πJ|ω0|1superscriptsubscript𝜇112𝜋𝐽superscriptsubscript𝜔01\mu_{1}^{1}=2\pi J|\omega_{0}|^{-1} (5.7)

which does not contain much information about the rod elements Ω1(h),,ΩJ(h)subscriptΩ1subscriptΩ𝐽\Omega_{1}(h),...,\Omega_{J}(h) of the junction Ξ(h)Ξ\Xi(h), namely only their number J𝐽J.

In order to construct the second term μ21superscriptsubscript𝜇21\mu_{2}^{1}, we, first of all, observe that a solution of the problem (5.5) can be subject to the orthogonality condition

ω0v(1)01(y)𝑑y=0subscriptsubscript𝜔0superscriptsubscript𝑣101𝑦differential-d𝑦0\int_{\omega_{0}}v_{(1)0}^{1}(y)dy=0 (5.8)

and, thus, formulas (5.5), (5.6) and (2.37), (2.39) lead to the representation

v(1)1(y)=J1|ω0|μ11(𝐆1(y)++𝐆J(y))=2π(𝐆1(y)++𝐆J(y))superscriptsubscript𝑣11𝑦superscript𝐽1subscript𝜔0superscriptsubscript𝜇11superscript𝐆1𝑦superscript𝐆𝐽𝑦2𝜋superscript𝐆1𝑦superscript𝐆𝐽𝑦v_{(1)}^{1}(y)=J^{-1}|\omega_{0}|\mu_{1}^{1}(\mathbf{G}^{1}(y)+...+\mathbf{G}^{J}(y))=2\pi(\mathbf{G}^{1}(y)+...+\mathbf{G}^{J}(y))

so that

+v(1)1=2π𝒢ε,+′′v(1)1=2π𝒬ε.formulae-sequencesuperscriptsubscriptWeierstrass-psuperscriptsubscript𝑣112𝜋𝒢𝜀superscriptsubscriptWeierstrass-p′′superscriptsubscript𝑣112𝜋𝒬𝜀\wp_{+}^{\prime}v_{(1)}^{1}=2\pi\mathcal{G}\varepsilon,\ \ \ \wp_{+}^{\prime\prime}v_{(1)}^{1}=-2\pi\mathcal{Q}\varepsilon. (5.9)

Owing to (5.8), the compatibility condition in the problem

Δyv(2)01(y)=μ21v(0)01(y)+μ11v(1)01(y),yω,zv(2)01(y)=0,yω0,formulae-sequencesubscriptΔ𝑦superscriptsubscript𝑣201𝑦superscriptsubscript𝜇21superscriptsubscript𝑣001𝑦superscriptsubscript𝜇11superscriptsubscript𝑣101𝑦formulae-sequence𝑦subscript𝜔direct-productformulae-sequencesubscript𝑧superscriptsubscript𝑣201𝑦0𝑦subscript𝜔0-\Delta_{y}v_{(2)0}^{1}(y)=\mu_{2}^{1}v_{(0)0}^{1}(y)+\mu_{1}^{1}v_{(1)0}^{1}(y),\ \ y\in\omega_{\odot},\ \ \ \ \partial_{z}v_{(2)0}^{1}(y)=0,\ \ y\in\partial\omega_{0},

reads

μ21|ω0|=(Δyv(2)01,v(0)01)ω0=v(2)1,+v(0)1superscriptsubscript𝜇21subscript𝜔0subscriptsubscriptΔ𝑦superscriptsubscript𝑣201superscriptsubscript𝑣001subscript𝜔0superscriptsubscriptWeierstrass-psuperscriptsubscript𝑣21superscriptsubscriptWeierstrass-psuperscriptsubscript𝑣01\mu_{2}^{1}|\omega_{0}|=-(\Delta_{y}v_{(2)0}^{1},v_{(0)0}^{1})_{\omega_{0}}=-\left\langle\wp_{-}^{\prime}v_{(2)}^{1},\wp_{+}^{\prime}v_{(0)}^{1}\right\rangle

and implies

μ21=2π|ω0|1(Clog+2π𝒢+2π𝒬)ε,ε.superscriptsubscript𝜇212𝜋superscriptsubscript𝜔01subscript𝐶2𝜋𝒢2𝜋𝒬𝜀𝜀\mu_{2}^{1}=2\pi|\omega_{0}|^{-1}\left\langle(C_{\log}+2\pi\mathcal{G+}2\pi\mathcal{Q})\varepsilon,\varepsilon\right\rangle. (5.10)

The diagonal matrices 𝒬𝒬\mathcal{Q} and Clogsubscript𝐶C_{\log} depend on length ljsubscript𝑙𝑗l_{j} and the reduced cross-section ωjsubscript𝜔𝑗\omega_{j} of the rod Ωj(h),subscriptΩ𝑗\Omega_{j}(h), j=1,,J𝑗1𝐽j=1,...,J, while the matrix 𝒢𝒢\mathcal{G} reflects disposition of the junction points P1,,PJsuperscript𝑃1superscript𝑃𝐽P^{1},...,P^{J} in the base of the plate Ω0(h)subscriptΩ0\Omega_{0}(h).

The terms (5.7) and (5.10) detached in (5.1) have been computed. An estimate of the asymptotic remainder μ~1(h)superscript~𝜇1\widetilde{\mu}^{1}(h) is a simple algebraic task and Theorem 16 converts the estimate into

|λ1(h)|lnh|1μ11|lnh|2μ21|c1|lnh|3.superscript𝜆1superscript1superscriptsubscript𝜇11superscript2superscriptsubscript𝜇21subscript𝑐1superscript3|\lambda^{1}(h)-|\ln h|^{-1}\mu_{1}^{1}-|\ln h|^{-2}\mu_{2}^{1}|\leq c_{1}|\ln h|^{-3}. (5.11)

5.2 The homogeneous junction

By setting in (1.13) the restrictions (1.15), the problem (1.6)-(1.12) reduces to the mixed boundary-value problem

Δxu(h,x)subscriptΔ𝑥𝑢𝑥\displaystyle-\Delta_{x}u(h,x) =λ(h)u(h,x),xΞ(h),formulae-sequenceabsent𝜆𝑢𝑥𝑥subscriptΞsquare-union\displaystyle=\lambda(h)u(h,x),\ \ \ x\in\Xi_{\sqcup}(h), (5.12)
u(h,x)𝑢𝑥\displaystyle u(h,x) =0,xΓ(h),νu(h,x)=0,xΞ(h)Γ(h),formulae-sequenceabsent0formulae-sequence𝑥Γformulae-sequencesubscript𝜈𝑢𝑥0𝑥subscriptΞsquare-unionΓ\displaystyle=0,\ x\in\Gamma(h),\ \ \ \partial_{\nu}u(h,x)=0,\ x\in\Xi_{\sqcup}(h)\setminus\Gamma(h),

stated in the intact domain Ξ(h)=Ω0(h)Ω1(h)ΩJ(h)subscriptΞsquare-unionsubscriptΩ0subscriptΩ1subscriptΩ𝐽\Xi_{\sqcup}(h)=\Omega_{0}(h)\cup\Omega_{1}(h)\cup...\cup\Omega_{J}(h), cf. (1.4) and (1.1), (1.3). An asymptotic analysis of the stationary problem (5.12) with a source term f(h,)L2(Ξ(h))𝑓superscript𝐿2subscriptΞsquare-unionf(h,\cdot)\in L^{2}(\Xi_{\sqcup}(h)) instead of λ(h)u(h,)𝜆𝑢\lambda(h)u(h,\cdot) has been developed in [4, Sect. 2]. Let us outline certain peculiarities of asymptotic models in the case (1.15), which are to be derived along the same scheme as in Sections 3 and 4.

As was mentioned in Section 1.2, a specific feature of the homogeneous junction Ξ(h)subscriptΞsquare-union\Xi_{\sqcup}(h) is that the limit operator decouples, see (5.17), i.e., instead of the connected skeleton in fig. 2,a, we obtain in the limit the disjoint domain ω2𝜔superscript2\omega\subset\mathbb{R}^{2} and intervals I1,,IJsubscript𝐼1subscript𝐼𝐽I_{1},...,I_{J}\subset\mathbb{R} as drawn in fig. 2,b.

Both the asymptotic models can be applied for the stationary problem of type (1.6)-(1.12) with the source term fpsubscript𝑓𝑝f_{p} instead of λup𝜆subscript𝑢𝑝\lambda u_{p} on the right-hand side of the Poisson equations. However, a result in [4] displays an explicit rational dependence on |lnh||\ln h| of the corresponding solution that furnishes its complete asymptotic form in finite steps. Moreover, it reduces an evident importance of the models for the spectral problem (1.6)-(1.12) where an argument based on the big parameter |lnh||\ln h| in (3.13) and (2.28) detects the holomorphic dependence on |lnh|1superscript1|\ln h|^{-1} which clearly cannot be described in finite steps.

Trying to formulate point conditions connecting the limit problems (1.20), (1.22) and (1.21), (1.23) in the skeleton Ξ0superscriptΞ0\Xi^{0} of the junction Ξ(h)Ξ\Xi(h), see fig. 1 and 2, we need a special solution of the homogeneous Neumann problem for the Laplace equation in the union Υj=ΛQjsubscriptΥ𝑗Λsubscript𝑄𝑗\Upsilon_{j}=\Lambda\cup Q_{j} of a layer and a semi-infinite cylinder. As was shown in [4], this solution gets the asymptotic behavior

W(ξ)𝑊𝜉\displaystyle W(\xi) =12π(ln1|η|+lnclog(ΛQj))+i=1,2Ki(ΛQj)ηj|η|2+O(|η|2),ξΛ,|η|+,formulae-sequenceabsent12𝜋1𝜂subscript𝑐Λsubscript𝑄𝑗subscript𝑖12subscript𝐾𝑖Λsubscript𝑄𝑗subscript𝜂𝑗superscript𝜂2𝑂superscript𝜂2formulae-sequence𝜉Λ𝜂\displaystyle=\frac{1}{2\pi}\left(\ln\frac{1}{|\eta|}+\ln c_{\log}(\Lambda\cup Q_{j})\right)+\sum\limits_{i=1,2}K_{i}(\Lambda\cup Q_{j})\frac{\eta_{j}}{|\eta|^{2}}+O(|\eta|^{-2}),\ \xi\in\Lambda,\ |\eta|\rightarrow+\infty, (5.13)
W(ξ)𝑊𝜉\displaystyle W(\xi) =|ωj|1ζ+O(eδζ),δ>0,ξQj,ζ+.formulae-sequenceabsentsuperscriptsubscript𝜔𝑗1𝜁𝑂superscript𝑒𝛿𝜁formulae-sequence𝛿0formulae-sequence𝜉subscript𝑄𝑗𝜁\displaystyle=|\omega_{j}|^{-1}\zeta+O(e^{-\delta\zeta}),\ \delta>0,\ \xi\in Q_{j},\ \zeta\rightarrow+\infty. (5.14)

It should be noticed that a proper choice of the coordinate origin eliminates the constants Ki=Ki(ΛQj)subscript𝐾𝑖subscript𝐾𝑖Λsubscript𝑄𝑗K_{i}=K_{i}(\Lambda\cup Q_{j}) because the change ηη=η+Kmaps-to𝜂superscript𝜂𝜂𝐾\eta\mapsto\eta^{\prime}=\eta+K with K2𝐾superscript2K\in\mathbb{R}^{2} provides the formulas

ln|η|𝜂\displaystyle\ln|\eta| =ln|ηK|=ln|η||η|2(Kη)+O(|η|3),absentsuperscript𝜂𝐾superscript𝜂superscriptsuperscript𝜂2𝐾superscript𝜂𝑂superscriptsuperscript𝜂3\displaystyle=\ln|\eta^{\prime}-K|=\ln|\eta^{\prime}|-|\eta^{\prime}|^{-2}(K\cdot\eta^{\prime})+O(|\eta^{\prime}|^{-3}),
|η|2ηisuperscript𝜂2subscript𝜂𝑖\displaystyle|\eta|^{-2}\eta_{i} =|η|2ηi+O(|η|3).absentsuperscriptsuperscript𝜂2superscriptsubscript𝜂𝑖𝑂superscriptsuperscript𝜂3\displaystyle=|\eta^{\prime}|^{-2}\eta_{i}^{\prime}+O(|\eta^{\prime}|^{-3}).

In what follows we assume that the points Pjsuperscript𝑃𝑗P^{j} and, therefore, the coordinates yj=hηjsuperscript𝑦𝑗superscript𝜂𝑗y^{j}=h\eta^{j} are fixed such that Ki(ΛQj)=0,subscript𝐾𝑖Λsubscript𝑄𝑗0K_{i}(\Lambda\cup Q_{j})=0, i=1,2,𝑖12i=1,2, in (5.13).

Repeating the matching procedure performed in Section 3.2, we keep the expansions (3.7), (3.8) but, according to (5.13), (5.14), replace (3.9), (3.10) with the following ones:

cj0+cj1Wj(ξ)superscriptsubscript𝑐𝑗0superscriptsubscript𝑐𝑗1subscript𝑊𝑗𝜉\displaystyle c_{j}^{0}+c_{j}^{1}W_{j}(\xi) =cj112πln1|ηj|+cj112πlnclog(ΛQj))+cj0+ in Ω(h),\displaystyle=c_{j}^{1}\frac{1}{2\pi}\ln\frac{1}{|\eta^{j}|}+c_{j}^{1}\frac{1}{2\pi}\ln c_{\log}(\Lambda\cup Q_{j}))+c_{j}^{0}+...\text{ \ \ in }\Omega_{\bullet}(h),
cj0+cj1Wj(ξ)superscriptsubscript𝑐𝑗0superscriptsubscript𝑐𝑗1subscript𝑊𝑗𝜉\displaystyle c_{j}^{0}+c_{j}^{1}W_{j}(\xi) =cj0+cj1|ωj|1ζ+ in Ωj(h).absentsuperscriptsubscript𝑐𝑗0superscriptsubscript𝑐𝑗1superscriptsubscript𝜔𝑗1𝜁 in subscriptΩ𝑗\displaystyle=c_{j}^{0}+c_{j}^{1}|\omega_{j}|^{-1}\zeta+...\text{ \ \ in }\Omega_{j}(h).

Thus, we obtain the relations

bjhh|ωj|1zvjh(0)=0,v^0h(Pj)vjh(0)=bjh(2π)1(lnh+lnclog(ΛQj))formulae-sequencesuperscriptsubscript𝑏𝑗superscriptsubscript𝜔𝑗1subscript𝑧superscriptsubscript𝑣𝑗00superscriptsubscript^𝑣0superscript𝑃𝑗superscriptsubscript𝑣𝑗0superscriptsubscript𝑏𝑗superscript2𝜋1subscript𝑐Λsubscript𝑄𝑗b_{j}^{h}-h|\omega_{j}|^{-1}\partial_{z}v_{j}^{h}(0)=0,\ \ \ \widehat{v}_{0}^{h}(P^{j})-v_{j}^{h}(0)=b_{j}^{h}(2\pi)^{-1}(\ln h+\ln c_{\log}(\Lambda\cup Q_{j})) (5.15)

which look quite similar to (3.12) but we have the small factor hh on the derivatives zvjh(0)subscript𝑧superscriptsubscript𝑣𝑗0\partial_{z}v_{j}^{h}(0), that is, on the projection ′′vsuperscriptsubscriptWeierstrass-p′′𝑣\wp_{-}^{\prime\prime}v. To perform the correct limit passage h+00h\rightarrow+0, we recall our previous asymptotic analysis in [4, Sect. 2] and make the substitution

vh=(v0h,v1h,,vJh)𝐯h=(𝐯0h,𝐯1h,,𝐯Jh)=(v0h,h1/2v1h,,h1/2vJh).superscript𝑣superscriptsubscript𝑣0superscriptsubscript𝑣1superscriptsubscript𝑣𝐽superscript𝐯superscriptsubscript𝐯0superscriptsubscript𝐯1superscriptsubscript𝐯𝐽superscriptsubscript𝑣0superscript12superscriptsubscript𝑣1superscript12superscriptsubscript𝑣𝐽v^{h}=(v_{0}^{h},v_{1}^{h},...,v_{J}^{h})\Rightarrow\mathbf{v}^{h}=(\mathbf{v}_{0}^{h},\mathbf{v}_{1}^{h},...,\mathbf{v}_{J}^{h})=(v_{0}^{h},h^{-1/2}v_{1}^{h},...,h^{-1/2}v_{J}^{h}). (5.16)

As a result, we conclude the point conditions

bj=0,vj0(0)=0,j=1,,J𝐯0=0J,+′′𝐯0=0Jformulae-sequenceformulae-sequencesubscript𝑏𝑗0formulae-sequencesuperscriptsubscript𝑣𝑗000formulae-sequence𝑗1𝐽superscriptsubscriptWeierstrass-psuperscript𝐯00superscript𝐽superscriptsubscriptWeierstrass-p′′superscript𝐯00superscript𝐽b_{j}=0,\ v_{j}^{0}(0)=0,\ j=1,...,J\ \ \ \ \Leftrightarrow\ \ \ \ \wp_{-}^{\prime}\mathbf{v}^{0}=0\in\mathbb{R}^{J},\ \wp_{+}^{\prime\prime}\mathbf{v}^{0}=0\in\mathbb{R}^{J} (5.17)

corresponding to the self-adjoint extension described in Remark 6. In other words, the limit spectrum (4.9) is composed from the spectrum {κh0}nsubscriptsuperscriptsubscript𝜅0𝑛\{\kappa_{h}^{0}\}_{n\in\mathbb{N}} of the Neumann problem in ω0subscript𝜔0\omega_{0} and the spectra {κhj=π2lj2n2}nsubscriptsuperscriptsubscript𝜅𝑗superscript𝜋2superscriptsubscript𝑙𝑗2superscript𝑛2𝑛\{\kappa_{h}^{j}=\pi^{2}l_{j}^{-2}n^{2}\}_{n\in\mathbb{N}} of the Dirichlet problems in Ij,subscript𝐼𝑗I_{j}, j=1,,J𝑗1𝐽j=1,...,J . It is worth to mention that, as was observed in [4], the substitution (5.16) has a clear physical reason, namely the energy functional for the problem (5.12) gets an appropriate approximation by the sum of the energy functionals for the above mentioned limit problems multiplied with the common factor hh.

Let us construct the first correction term in the asymptotics

λ1(h)=0+hμ11+λ~1(h)superscript𝜆10superscriptsubscript𝜇11superscript~𝜆1\lambda^{1}(h)=0+h\mu_{1}^{1}+\widetilde{\lambda}^{1}(h) (5.18)

of the first eigenvalue of the problem (5.12). Recalling an asymptotic procedure in [4, Section 2], we search for the corresponding eigenfunction in the form

uh(h,x)superscript𝑢𝑥\displaystyle u^{h}(h,x) =1+hv(1)01(x)+in Ω(h),absent1superscriptsubscript𝑣101𝑥in subscriptΩ\displaystyle=1+hv_{(1)0}^{1}(x)+...\ \ \ \text{in }\Omega_{\bullet}(h), (5.19)
uh(h,x)superscript𝑢𝑥\displaystyle u^{h}(h,x) =1lj1z+in Ωj(h).absent1superscriptsubscript𝑙𝑗1𝑧in subscriptΩ𝑗\displaystyle=1-l_{j}^{-1}z+...\ \ \ \text{in }\Omega_{j}(h).

Regarding (5.19) as outer expansions, we write the inner expansions in the vicinity of the sockets θjhsuperscriptsubscript𝜃𝑗\theta_{j}^{h} as follows:

uh(h,x)=1hlj1|ωj|Wj(h1(yPj),h1z)+superscript𝑢𝑥1superscriptsubscript𝑙𝑗1subscript𝜔𝑗subscript𝑊𝑗superscript1𝑦superscript𝑃𝑗superscript1𝑧u^{h}(h,x)=1-hl_{j}^{-1}|\omega_{j}|W_{j}(h^{-1}(y-P^{j}),h^{-1}z)+...

Finally, we take the representations (5.13), (5.14) and apply the matching procedure, cf. Section 3.2, to close the problem (5.5) with the asymptotic conditions near the points P1,,PJsuperscript𝑃1superscript𝑃𝐽P^{1},...,P^{J}

v(1)01(x)=12πjχj(y)|ωj|ljln1rj+v^(1)01(x),v^(1)01H2(ω0).formulae-sequencesuperscriptsubscript𝑣101𝑥12𝜋subscript𝑗subscript𝜒𝑗𝑦subscript𝜔𝑗subscript𝑙𝑗1subscript𝑟𝑗superscriptsubscript^𝑣101𝑥superscriptsubscript^𝑣101superscript𝐻2subscript𝜔0v_{(1)0}^{1}(x)=-\frac{1}{2\pi}\sum\nolimits_{j}\chi_{j}(y)\frac{|\omega_{j}|}{l_{j}}\ln\frac{1}{r_{j}}+\widehat{v}_{(1)0}^{1}(x),\ \ \ \widehat{v}_{(1)0}^{1}\in H^{2}(\omega_{0}). (5.20)

Similarly to Section 5.1 the compatibility condition in the problem (5.5), (5.20) converts into the formula

μ11=|ω0|1(|ω1|l11++|ωJ|lJ1).superscriptsubscript𝜇11superscriptsubscript𝜔01subscript𝜔1superscriptsubscript𝑙11subscript𝜔𝐽superscriptsubscript𝑙𝐽1\mu_{1}^{1}=|\omega_{0}|^{-1}(|\omega_{1}|l_{1}^{-1}+...+|\omega_{J}|l_{J}^{-1}).

Combining approaches in Section 4 and [4, Sect. 2], the asymptotic formula (5.18) can be justified by means of the estimate |λ~1(h)|c1h2(1+|lnh|)superscript~𝜆1subscript𝑐1superscript21|\widetilde{\lambda}^{1}(h)|\leq c_{1}h^{2}(1+|\ln h|) for the remainder.

Let us consider the problems (1.20), (1.22) and (1.21), (1.23) connected through the point conditions

𝐯h+h1/2′′𝐯h=0,h1/2+′′𝐯h+𝐯hSh𝐯h=0J,formulae-sequencesuperscriptsubscriptWeierstrass-psuperscript𝐯superscript12superscriptsubscriptWeierstrass-p′′superscript𝐯0superscript12superscriptsubscriptWeierstrass-p′′superscript𝐯superscriptsubscriptWeierstrass-psuperscript𝐯superscript𝑆superscriptsubscriptWeierstrass-psuperscript𝐯0superscript𝐽\wp_{-}^{\prime}\mathbf{v}^{h}+h^{1/2}\wp_{-}^{\prime\prime}\mathbf{v}^{h}=0,\ \ \ h^{-1/2}\wp_{+}^{\prime\prime}\mathbf{v}^{h}-\wp_{+}^{\prime}\mathbf{v}^{h}-S^{h}\wp_{-}^{\prime}\mathbf{v}^{h}=0\in\mathbb{R}^{J}, (5.21)

where γj=ρj=1subscript𝛾𝑗subscript𝜌𝑗1\gamma_{j}=\rho_{j}=1 because the junction is homogeneous and the J×J𝐽𝐽J\times J-matrix Shsuperscript𝑆S^{h} is given by formula (3.13) with clog(ΛQj)subscript𝑐Λsubscript𝑄𝑗c_{\log}(\Lambda\cup Q_{j}) instead of clog(ωj)subscript𝑐subscript𝜔𝑗c_{\log}(\omega_{j}), see (5.13) and (3.4).

The conditions (5.21) follow immediately from (5.15) after substitution (5.16). They are involved into the symmetric generalized Green formula of type (2.35)

q(vh,wh)𝑞superscript𝑣superscript𝑤\displaystyle q(v^{h},w^{h}) =+vhh1/2+′′vh+Shvh,whvh,+whh1/2+′′wh+ShwhabsentsuperscriptsubscriptWeierstrass-psuperscript𝑣superscript12superscriptsubscriptWeierstrass-p′′superscript𝑣superscript𝑆superscriptsubscriptWeierstrass-psuperscript𝑣superscriptsubscriptWeierstrass-psuperscript𝑤superscriptsubscriptWeierstrass-psuperscript𝑣superscriptsubscriptWeierstrass-psuperscript𝑤superscript12superscriptsubscriptWeierstrass-p′′superscript𝑤superscript𝑆superscriptsubscriptWeierstrass-psuperscript𝑤\displaystyle=\left\langle\wp_{+}^{\prime}v^{h}-h^{-1/2}\wp_{+}^{\prime\prime}v^{h}+S^{h}\wp_{-}^{\prime}v^{h},\wp_{-}^{\prime}w^{h}\right\rangle-\left\langle\wp_{-}^{\prime}v^{h},\wp_{+}^{\prime}w^{h}-h^{-1/2}\wp_{+}^{\prime\prime}w^{h}+S^{h}\wp_{-}^{\prime}w^{h}\right\rangle
+h1/2+′′vh,wh+h1/2′′whh1/2vh+h1/2′′vh,+′′wh,superscript12superscriptsubscriptWeierstrass-p′′superscript𝑣superscriptsubscriptWeierstrass-psuperscript𝑤superscript12superscriptsubscriptWeierstrass-p′′superscript𝑤superscript12superscriptsubscriptWeierstrass-psuperscript𝑣superscript12superscriptsubscriptWeierstrass-p′′superscript𝑣superscriptsubscriptWeierstrass-p′′superscript𝑤\displaystyle+h^{-1/2}\left\langle\wp_{+}^{\prime\prime}v^{h},\wp_{-}^{\prime}w^{h}+h^{1/2}\wp_{-}^{\prime\prime}w^{h}\right\rangle-h^{-1/2}\left\langle\wp_{-}^{\prime}v^{h}+h^{1/2}\wp_{-}^{\prime\prime}v^{h},\wp_{+}^{\prime\prime}w^{h}\right\rangle,

and therefore, all requirements in Section 2 and 3 with slight modifications apply to the model in the case (1.15), too. Moreover, a simple analysis requiring only for algebraic operations as in Section 5.1, provides the representation

μ1(h)=hμ11+O(h2(1+|lnh|))superscript𝜇1superscriptsubscript𝜇11𝑂superscript21\mu^{1}(h)=h\mu_{1}^{1}+O(h^{2}(1+|\ln h|)) (5.22)

of an eigenvalue in the model (1.20)-(1.23), (5.21) which is supplied with an operator of type (2.33) on the function space (2.13) with detached asymptotics.


Acknowledgements.


The work of S.A.N. was supported by grant 15-01-02175 of Russian Foundation for Basic Research. G.C. is a member of GNAMPA of INDAM.


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